•.     <       - 
*  .^?vi    ? 

•  -! .  •  - ;-' 


V 


r 


• 


SCHOOLMASTER'S  ASSISTANT: 

IMPROVED  AND  ENLARGED, 

BEING  A*PLAI»        » 

PRACTICAL  SYSTEM 

** 

«• 
OF 

• 

ARITHMETIC  :*  . 

% 

ADAPTED   TO    THEJNITED    RTATEsT 


EDITION. 


BY  NATHAN  DABOLL. 


»RH1TED  AND  PrBLISHEB  BY  SAMUEL  GREEN,  FROFftlBTOfc 
OF    THE    COPT    RIGHT. 


C  Siarr,  Stereotype  Founder*  A*.  T, 


DISTRICT  OF   CONNECTICUT,  SS.j 

t  g  BE  IT  REMEMBERED,  That  on  the  twentr- 
first  day  of  October,  in  the  thirty -sixth  year  of  the 
Independence  of  the  United  States  of  America,  SAMUEL 
GREKN,  of  said  District,  hath  deposited  in  tWs  office  the 
title  of  a  Book,  the  right  whereof  he  claims  al  proprie- 
tor, in  the  words  folUjiving,  to  wit : — "  Daboll's  School- 
master's Assistant  T  improved  and  enlarged.  Being  a 
plain  practical  system  of  Arithmetic  :  adapted  to  the 
United  Sfates.  Stereotype  Edition.  By  NATHAN  DA- 
BOI.L."  ."  0 

In  conformity  to  the  Act  of  the  Congress  of  the  United 
States,  entitle9, "  An  Act  for  tMe  encouragement  of  learn- 
ing, by  securing  the  copi$y  of  Maps,  Charts  and  Books, 
to  the  Authors  and  proprietors  of  them  during  the  time* 
therein  mentioned." 

HENRY  W.  EDWARDS, 
Clerk  of  the  District  of  Connecticut. 
A  true  copy  of  Record :   Examined  and  sealed  by  me, 
H.  W.  EDWARDS,  Clerk  ofthe'JHst.  of  Conn. 


RECOMMENDATIONS. 


YALE-COLLEGE,   »OV.   27,    1799. 

1  HAVTS  read  DABOLL'S  SCHOOLMASTER'S  ASSISTANT. 
The  arrangement  of  the  different  branches  of  Arithmetic 
is  judicious^md  perspicuous.  The  author  has  Well  ex- 
plained Decimal  Arithmetic,  and  has  applied  it  in  a  plain 
and  elega'nt  manner  in  the  solution  of  various  questions, 
und  especially  to  those  relative  to  the  Federal  Conjputa- 
tion  of  money.  I  think  it  will  be  a  very  useful  book  to 
Schoolmasters  and  their  pupils. 

JOS1AH  MEIG$,  Professor  of 
Mathematics  and  Natural  Philosophy* 
[Now  Surveyor  General  of  the  United  States.] 

1  HAVE  given  some  attention  to  the  work  above  men- 
tioned, and  concur  with  Mr.  Professor  Meigs  in  his  opin- 
ion of  its  merit.  NOAH  WEBSTER. 
New-Haven,  December  1£,  1799. 

^  RHODE^|p££ND    COLLEGE,    NOV.   SO,    1799. 

I  HAVE  run  through  Mr.  DABOLL'S  SCHOOLMASTER'S 
ASSISTANT,  and  have  formed  of  it  a  very  favourable  opin- 
ion. According  to  its  original  design,  I  think  it  well 
"  calculated  to  furnish  Schools  in  general  with  a  method- 
ical, easy  and  comprehensive  System  of  Practical  Arith- 
metic." I  therefore  hope  it  may  find  a  generous  patron- 
age, and  have  an  extensive  spread. 

ASA  MESSER,  Professor  of  the 
teamed  Languages,  and  Teacher  of  Mathematics 
\  Now  President  of  that  Institution."] 


2030032    , 


RECOMMENDATIONS. 
FLAISJFIELD    ACADEMY,    APRIL   20,    1802. 

I  MAKE  use  of  DABOLL'S  SCHOOLMASTER'S  ASSISTANT, 
in  teaching  common  Arithmetic,  and  think  it  the  best 
calculated  for  that  purpose  of  any  which  has  fallen  within 
my  observation,  JOHN  ADAMS,  Rector  of 

Plttinfield  Academy 
{  [Now  Principal  of  Phillips  Academy,  Andover,  Mass.  J 

BlLLERICA    ACADEMY,  (MASS.)   DEO.    10,    1-807. 

HAVING  examined  Mr.  DABOLL'S  System  of  Arith- 
metic, I  am  pleased  with  the  judgment  displayed  in  his 
method,  and  the  perspicuity  of  his  explanations,  and 
thinking  it  as  easy  and  comprehensive  a  system  as  any 
•with  which  I  am  acquainted,  can  cheerfully  recommend 
it  to  the  patronage  of  Instructors. 

SAMUEL  WHITING, 
Teacher  of  Mathematics. 

TROM  MR.  KENNEDY,  TEACHER  OF  MATHEMATICS. 

*• 

I  BECAME  acquainted  with  DABOLL'S  SCHOOLMAS- 
TWt's  ASSISTANT,  in  the  year  1802.  and  on  examining  it 
attentively,  gave  it  my  decided  preference  to  any  other 
•ystem  extant,  and  immediately  adopted  it  for  the  pupils 
under  my  charge  ;  and  since  that  time  have  used  it  exclu- 
sively in  eJementary  tuition,  to  the  great  advantage  and 
improvement  of  the  student,  as  well  as  the  ease  and  as- 
sistance of  the  Preceptor.  I  also  deem  it  equally  well 
calculated  for  the  benelit  of  individuals  in  private  in- 
struction ;  and  think  it  my  duty  to  give  the  labour  and 
ingenuity  of  the  author  the  tribute  of  my  hearty  approval 
and  recommendation. 

ROGFK  KENNEDY, 
New-York,  March  £0,  1811, 


PREFACE. 


1  HE  design  of  this  work  is  to  furnish  the  schools  of 
the  United  States  with  a  methodical  and  comprehei)sive 
system  of  Practical  .arithmetic,  in  «  hich  I  have  endea- 
voured, through  the  whole,  to  have  the  rules  as  concise 
and  familiar,  as  the  nature  of  the  subject  will  permit. 

During  the  Jong  period  which  I  have  devoted  to  the 
instruction  of  youth  in  Arithmetic,  I  have  made  use  ot 
various  systems  which  have  just  claims  to  scientific  mer- 
it ;  but  the  authors  appear  to  have  been  deficient  in  an 
important  point — the  practical  teacher's  experience. — 
They  have  been  too  sparing  of  examples,  especially  in 
the  first  rudiments ;  in  consequence  of  which,  the  young 
pupil  is  hurried  through  the  ground  rules  too  fast  for  his 
capacity.  This  objection  I  have  endeavoured  te  obviatt 
in  the  following  treatise. 

In  teaching  the  first  rules,  I  have  found  it  best  to  en- 
courage the  attention  of  scholars  by  a  variety  of  easy  ami 
familiar  questions,  which  might  serve  to  strengthen  thei; 
mim:s  as  their  studies  grow  more  arduous. 

The  rules  are  arranged  in  such-  order  as  to  introduce 
the  most  simple  and  necessary  parts,  previous  to  those 
which  are  more  abstruse  and  difficult. 

To  enter  into  a  detail  of  the  whole  work  would  be  <«- 
dious  ;  I  shall  therefore  notice  onlj  a  few  particulars,  anti 
refer  the  reader  to  the  contents. 

Although  the  Federal  Coin  is  purely  decimal,  it  is  f* 
nearly  allied  to  whole  numbers,  and  so  absolutely  neces- 
sary to  be  understood  by  every  one,  that  I  have  intro- 
duced it  immediately  after  addition  of  whole  number-,, 
and  also  shown  how  to  find  t!.c  value  of  gotx! 
immediately  after  simple  multiplication;  which  may  b« 
of  great  advantage  to  many,  who  perhaps  will  not  hav« 
an  opportunity  of  learning  fractions. 

In  the  arrangement  of  fractions,  I  have  taken  an  entire 
new  method,  the  advantages  and  facility  of  which  wili 
sufficiently  apologize  for  its  not  beii^  rarirtfcatg  to  othar 


VI  PREFACE. 

s  y  stems.  As  decimal  fractions  may  be  learned  much  easier 
than  vulgar,  and  are  more  simple,  useful,  and  neces- 
sary, and  soonest  wanted  in  more  useful  branches  of 
Arithmetic,  they  ought  to  be  learned  first,  and  Vulgar 
Fractions  omitted,  until  further  progress  in  the  science 
shall  make  them  necessary.  It  may  be  well  to  obtain  a 
general  idea  of  them,  and  to  attend  to  two  or  three  easy 
problems  therein :  after  which,  the  scholar  may  \earn 
decimals,  which  will  be  necessary  in  the  reduction  of  cur- 
rencies, computing  interest  and  many  other  branches. 

Besides,  to  obtain  a  thorough  knowledge  of  Vulgar 
Fractions,  it  generally  a  task  too  hard  for  young  scholars 
who  have  made  no  further  progress  in  Arithmetic  than 
Reduction,  and  often  discourages  them. 

I  have  therefore  placed  a  few  problems  in  Fractions, 
according  to  the  method  above  hinted ;  and  after  going 
through  the  principal  mercantile  rules,  have  treated  upou 
Vulgar  Fractions  at  large,  the  scholar  being  now  capable 
of  going  through  them  with  advantage  and  ease. 

In  Simple  Interest,  in  Federal  Money,  I  have  given 
several  new  and  concise  rules ;  some  of  which  are  par- 
ticularly designed  for  the  use  of  the  compting-house. 

The  Appendix  contains  a  variety  of  rules  for  casting 
Interest,  Rebate,  &c.  together  with  a  number  of  the  most 
easy  and  useful  problems,  for  measuring  superficies  and 
solids,  examples  of  forms  commonly  used  in  transacting 
business,  useful  tables,  &c.  which  are  desigaed  as  aids  in 
the  common  business  of  life. 

Perfect  accuracy,  in  a  work  of  this  nature,  can  hardly 
be  expected  ;  errors  of  the  press,  or  perhaps  of  the  au- 
thor, may  have  escaped  correction.  If  any  such  are  point- 
ed out,  it  will  be  considered  as  a  mark  of  friendship  and 
favor,  by 

The  public's  most  humble 

and  obedient  Servant, 

NATHAN  DABOLfc. 


TABLE  0V  CONTENTS. 

Pagt. 

ADDITION,  simple      .      .  17 

of  Federal  Money     ...  22 

Compound            ....  40 

Alligation        .                 189 

Annuities  or  Pensions,  at  Compound  Interest      .  2.P.5 

Arithmetical  progression           ....  194 

Barter 138 

Brokerage 125 

Characters,  Explanation  of          .         .         »         .  14 

Commission                        124 

Conjoined  Proportion           .....  149 

Coins  of  the  United  States,  Weights  of     .         .  £32 

Division  of  "NY hole  Numbers  S-* 

Contractions  in           ....  57 

Compound              57 

Discount                   135 

Duodecimals 228 

Equation  of  Payments             .         .         .        .  138 

Evolution,  or  Extraction  of  Roots        .        .       »  179 

Exchange ,         .         .  151 

Federal  Money           .  £! 

Subtraction  ot         .        „  27 

Fellowship                                                      .        .  i44 

Compound       ...                  .  14f> 

Fractions,  Yulgar  and  Decimal  .        .        74, 

Insurance        

Interest,  Simple 120 

—  by  Decimals          .  169 

Compound             .  134. 

by  Decimals           .         .         .  MT 


Inverse  Proportion 

Involution        .                  17  8 

;uid  Gain             140 

Multiplication-,  Simple £3 

Application  and  vse  of 

Supplimeut  to 

. Compound           ....  51 

Numeration     .......  15 

Practice  .         .         *         .         .         .         .109 

Position            .......  200 

Permutation  of  Quantifra*                             »  20.4 


\'y"l  TABLE   OF   CONTENTS/ 

Page. 

Questf  y  is  lor  exercise 209 

Reduction 63 

of  Currencies,  do.  of  Coin          .  89,  93 

Rule  of  Tliree  Direct,  do.  Inverse  .  100,  108 

Double .  148 

Rules,  for  reducing  the  different  currencies  of  the 
several  United  States,  also  Canada  and  No- 
va-Scotia, each  to  the  par  of  all  others  96,  97 

Application  ef  the  preceding     ...  98 

• Short  Practical,  for  calculating  Interest  126 

for  casting  Interest  at  6  per  cent.      .         .  215 

• for  finding  the  contents  of  Superfices  &  Solids    220 

to  reduce  the  currencies  of  the  different 

States,  to  Federal  Money          .        .         .  218 
Rebate,  A  short  method  of  nnding  the,  of  any  giv- 
en sum  for  months  and  days         .        .  217 

Subtraction,  Simple-            £5 

Compound           ....  45 

Table,  Numeration  and  Pence    ....  9 

Addition,  Subtraction,  and  Multiplication  10 

— — .  of  Weight  and  Measure            .        .        .  11 

of  Time  and  Motion     ....  13 

showing  the  number  of  days  from  any  day 

ftf  one  mouth,  to  the  same  day  in  any  other 

month 172 

"          showing  the  amount  of  ll.  or  1  dollar,  at  5  & 

6  per  cent.  Compound  Interest,  for  20  years  253 

showing  the  amount  of  ll.  annuity,  forborne 

for  31  years  or  under,  at  5  and  6  per  cent. 

Compound  Interest           .        .        .        .  335 

showing  the  present  worth  of  It.  annuity,  far 

31  yrs.  at  5  &  6 per  c.  Compound  Interest  ib. 

of  cents,  answering  to  the  currencies  of  the 

United  States,  with  Sterling,  &c.      .        .  £36 

showing  the  value  of  Federal  Money  in 

other  currencies        ...                 .  037 

Tare  and  Trett ^14 

Useful  Forms  in  transacting  business           .    "    .  233 
Weights  of  several  pieces  oi  English,  Portuguese, 

&  French,  gold  coins,  in  dollars,  cts.  &  mills  234 

ot  English  &  Portuguese  gold,      do.      do.  285 

of  French  and  Spanish  gold,         do.      do. 


DABOU/S 


SCHOOLMASTER'S  ASSISTANT 


ARITHMETICAL  TABLES. 


Numeration  TgbU 

P<pnc«  1 

(0k 

d.       8.  d. 

-. 

-e 

20  is  1     8 

3 

c 

VI 

on 

SO      2     6 

. 

g 

c: 

40      3    4 

i 

2 

«£ 

0* 

in 

50      4     2 

>*-< 

us 

1— 

8 
O 

60      5     0 

•o 
m 
ts 

i 

en 

C 
w 

•e 

P 

M 

T3 

c 

•3 

70      5  10 
80      6    8 

£ 
•a 

A 

•3   S 

w 

H  S 

a> 

•« 

c 

£ 

IT* 

0 

GB 

C 

H 

i 
1 

V 

fc. 

-a 

3 

tr 

1 

.•§ 
'S 

90      7     6 
100      8    4 
110      9    2 

,9 

8     7 

"5 

5 

4 

3 

2 

1 

120    10    0 

9    8 

7 

6 

5 

4 

3 

2 

9 

8 

7 

6 

5 

4 

3 

9 

8 

7 

6 

5 

4 

••^^ 

9 

8 

7 

6 

5 

9 

8 

7 

6 

mac 

9 

8 

7 

4  farthings  1 

9 

8 

12  pence,    1  s 

9 

20  shillings,  1  . 

d.  i 
12  Is  I 
24  £ 
S6 
43 
60 
72 
84 
96 
108 


S 
4 
5 
6 
7 
8 
9 

120     10 
132    11 


10  ARITHMETICAL   TABLKS. 

ADDITION   AND   SUBTRACTION  TABLE. 


1 

2 

s 

4 

5 

6 

7 

8 

9 

10 

11 

12  j 

2 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

3 

5 

6 

I 

8 

9 

10 

11 

12 

13 

14 

15 

4 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

5 

7 

8 

9 

JO 

11 

12 

IS 

14 

15 

16 

17 

^ 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18; 

7 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

8 

10 

H 

12 

\j 

14 

15 

16 

17 

18 

19 

ill) 

9 

11 

12 

13 

14 

la 

16 

17 

18 

19 

20 

21 

10 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

MULTIPLICATION     TABLE. 


1  1 

2 

3 

4 

5 

6 

7  :    8 

9 

10)  111  12 

2 

hr 

4 

6 

Q 

10 

12 

1-1 

16 

18 

20)  22|  24 

6 

9 

12 

15 

18 

21 

24 

27 

S0|  3S|  36j 

B 

8 

12 

16 

20 

2.4 

23 

32 

36 

40|  44)  48' 

0 

10 

15 

20 

25 

30 

35 

40 

4j 

50]  65  1  60 

6 

7 

12 

Lb 

24 

30 

42 

48 

0() 

CO  I  Goj  72 

ii 

2l 

28 

35 

70j  771  84 

24 

!S2 

40. 

o4 

72 

80|  88j  96 

IfO 

9~ 

36 

45 

8! 

OOJ  991108 

30 

40 

60 

TO 

80 

90 

IOOI110J120 

22 

,>  "S 

l>,'> 

44 

55 

66 

-7 

a8 

99 

110|12t|13C 

[IST]  24  |  3C 

48 

60 

r& 

84 

96 

10S 

120J132|1  -'4 

To  learn  this  Table 
fcand  column,  and  the 
common  angle  of  nice 
a.lon^  at  the  right  han< 
you  will  find  the  prodi 

Find  your  multiplier  in  the  left 
multiplicand  a-top,  and  in  the 
ting,  or  against  your  multiplier, 
1,  and  under  your  multiplicand, 
ict,  or  answer." 

ARITHMETICAL    TABLES.  1* 

2.  Troy  Weight. 

24  grains  (.gr.)  make    1  penny-weight,  marked       pwt, 
£0  penny -weights,        1  ounce,  oz. 

12  ounces,  1  pound,  Ib 

5.  Avoirdupois  Weight. 

16  drains  (dr.]  make  1  ounce,  oz. 

16  ounces,  1  pound,  Ib. 

28  pounds,  1  quarter  of  a  hundred  weight,  qr. 

4  quarters,  1  hundred  weight,  cwt. 

20  hundred  weight,  1  ton,  T. 

By  this  weight  are  weighed  all  coarse  and  drossv 
grocery  wares,  and  all  metals  except  gold  and  silver. 

4.  Apothecaries  Weight. 

20  grains  (gr.)  make  1  scruple,  9 

3  scruples,  1  dram,  5^ 

8  drams,  1  ounce,  3 

12  ounces,  1  pound,  ffc, 

Apothecaries  use  tms  weight  in  compounding  their 
medicines. 

5.  doth  Measure. 

4  nails  (na.)  make  1  quarter  of  a  yard,    qr. 

4  quarters,  1  yard,  yd. 

3  quarters,  I  fell  Flemish,  >, 

5  quarters,  1  Ell  English,  E.  E. 

6  quarters,  1  Ell  French,  E.Fr. 

6.  Dry  Msasure. 

Q  pints  (pt.}  make  1  quart,  qt, 

8  quarts,  1  p.  j,k. 

4  pecks,  1  bushel,  bu. 

This  measure  is  applied  to  grain,  beans,  flax -seed,  saU. 
oats,  oysters,  coali  &c 


.?•  ARITHMETICAL  TABLES. 

7.  Wine  Measure. 

'<*  (gi.)  make      1  pint,  ft. 

C  j.  1  quart,  qt. 

4  qi  .  1  pjallon,  gal. 

51}  1  barrel,  ll. 

1  tierce,  tier. 

1  hogshead)  hhd. 

2  hogsheads,  1  pipe,  p> 

1  tun,  T. 

All  brandies,  spirit^,  nif?:i'l.  vinegar,  oil,  &c.  are  mea- 
sured by  wine  measure.  +\'ote. — '231  solid  inches,  make 
a  gallon. 

8.  Long  Measure. 

leycorni  (ft.  c.)  make  1  inch,  marked          in. 
1  font,  ft. 

1  yard,  yd. 

1  rod,  pole,  or  perch,  rd. 
I  furlong,  fur. 

1  mile,  m. 

1  lf,i  lea. 

\  deujMN-,  on  tine  earth. 
*•  circumferem  <  irth. 

the  distance  of 
.^th  is  considered, 

-t  of  hordes,  4  inches  make 

In   mc^  -ke  1    lathoin, 

•  I  bj  a  chain, 
.-.  conuiuiug  one  hundred  links. 


ARITHMETICAL  TABLES.  1$ 

9.  Land,  or  Square  Measure. 

144  square  inches  maka  1  square  foot. 

9  square  feet,  1  square  yard. 
SOI  r  square  yards,  or  >  d 

2/2$  square  ieet,        5 

40  square  rods,  1  square  rood* 

4  square  roods,  1  square  acre. 

640  square  acres,  1  square  mile. 

10.  Solid  or  Cubic  Measure. 

1728  solid  inches  make  1  solid  foot 

40  feet  of  round  timber,  or?      .   . 

~n  f   4    c  i          ,-    ,  >      1  ton  or  load., 

oO  feet  ot  hewn  timber,        ^ 

128  solid  feet  or  8  feet  long,  >      1     ^  of  w(j()i]> 
4  wide,  and  4  high,  j 

All  solids,  or  things  that  have  length,  breadth  and  de]>-'», 
are  measured  by  this  measure.     N.  B.  The  wine  gaiiou 
contains  231  solid  or  cubic  inches,  and  the  beer  g; 
282.     A  bushel  contains  2150,42  solid  inches, 

1  1.     Time. 

60  seconds  (S.)  make       1  minute,  marked        ft.  _Y 
60  minutes,  1  hour, 

24  hours,  1  dav. 

7  days,  1   wi-rk, 

4  weeks,  1   month, 

13  months,  1  day  and  6  hours,  1  Julian  year,      yr. 

Thirty  days  hath  September,  April,  June,  and  Noveii 
February  twetity~eight  alone,  all  the  rest  have  ihirtv- 
N.  i?.  lu  bissextile,  or  leap  year,  Ftbnnt; 


T2.  circular  Motion. 

•uls  ('')  make  1  minute, 

us,  I   decree, 

1  si»n, 
1)  degrees,  the  whole  great 


CHARACTERS. 

Explanation  of  Characters  tised  in  thin  Hook. 


«*  Equal  to.  a>  !?</.  =  Is.  signifies  that  12  pence  are 
equal  to  I  shilling. 

-f  More,  the  sign  of  addition,  as  5-f  7=12,  signifies 
that  5  and  7  :umed  together,  are  equal  to  12. 

—  Minus,  or  less,  the  sign  of  subtraction,  as  6—2=4, 
signifies  that  2  subtracted  from  6,  leaves  4. 

X  Multiply,  or  with,  the  sign  of  Multiplication  ;  as 
4x3=12,  signifies  that  4  multiplied  by  3,  is  equal  t» 
12. 

•n  ;  as  8-7-2=4,  signifies  that  8 
aal   to  4  ;  or  thus,   |=»4,  each  of 
which  signify  the  same  thing. 

5  :  Four  points  set  in  the  middle  of  four  numbers,  denote 
them  to  '  :<mnl  to  <>;.e  another,  by  the  rule 

as  2  :  4  :  :  ii  :  i  G  j  that  is,  as  2  to"  4,  so  is  8 
to 

x/  Pif'ired  to  anv  number,  supposes  that  the  square  root 

number,*.  i:.e  cube  root  of  that 

ed. 


the  biquadrute  ronf,  at  ionrtli  po\vc»> 


15 

ARITHMETIC. 

ARITHMETIC  is  the  art  of  commuting  by  numbers, 
and  lias  five  principal  rules  for  its  operation,  viz.  Nume- 
ration, Addition,  Subtraction,  Multiplication,  and  Divi- 
sion. 

NUMERATION. 

Numeration  is  the  art  of  numbering.  /  It  teaches  to 
express  the  value  of  any  proposed  number  by  the  follow- 
ing characters,  or  figures  : 

1 ,  2,  5,  4,  5,  6,  7,  8,  9,  0 — or  cypher. 
'«*  the  simple  value  of  figures,  each  has  a  local 
value,  which  depends  upon  the  place  it  stands  in,  vi/.. 
any  figure  in  the  place  of  units ;  represents  only  its  sim- 
ple value,  or  so  many  ones,  but  in  the  second  place,  or 

NOTE. — Although  a  cypher  standing  alone  signifies  noth- 
ing ;  yet  when  it  is  placed  on  the  rij;ht  hand  of  figures,  it  in- 
ereases  their  value  in  a  tenfold  p  .  l>y  throwing  them 

into  higher  places.  Thus  2  with  a  cypher  annexed  to  it, 
becomes  20,  twenty,  and  with  two  cyphers,  thus,  £00,  two 
hundred 

2.  "When  numbers  consisting  of  m.iny  figures,  arc  given  to 
he  read,  it  will  be  found  convenient  to  divide  them  into  as 
many  periods 'as  we  can,  of  f>ix  figures  each,  reckonirg  from 
tho  ri^lit  hand  towards  the  left,  coi'm-  the  first  the  period  of 
units,  the.  second  that  of  millions,  the.  third  billions,  tile  fourth 
tiTtiions,  &tr..  as  in  the  following  number : 

0  7   S  6   £  5  4  6    2   7  JJ    9  0    I  2  5  0  6  7  9  2 


4.    Pciind  of  1  S.    rfr'od  of 
Trillions.      I       Billions. 


8073 


625462 


2.    Pf, 

ions. 


1.    Period  (f 

Ur 


The  foregoing  number  is  read  thus—  Ei; 
seventy-three  trillions;   six  hundred  and   twenty-five  thou- 
sand, four  hundred  and  sixty-hvo  hillions  ;  s^v- 
eighty-nine  thousand  and  twelve  i;i  idred  and 

six  thousand,  seven  hundred  and  ninety- f- 

N.  B.  Billions  is  substitute  \  for  millions  of  millions. 

Trillions  for  millions  of  mi-lions  of  millions. 

Qaatrillions  for  millions  of  millions  of  mil'iion«  of  r 
be. 


16 


NUMERATION. 


place  o!  tens,  it  becomes  so  many  tens,  or  ten  times  its 
simple  value,  and  in  the  third  place,  or  place  of  hundreds, 
*ht  becomes  an  hundred  times  its  simple  value,  and  so  on, 
as  in  the  following 

TABLE: 


98765 


-  One. 

-  Twenty-one. 

-  Three  fiu  rid  red  twenty-one. 

-  Four  thousand  521. 

-  Fifty-four  thousand  32*. 

-  654  th<Mttnd  321. 

-  7  million  054  thousand  321. 

,11'ion  t>54  thousand  321. 

-  UK7  million  654  thousand  321. 

-  123  million  456  thousand  789» 

-  987  million  654  thousand  348. 


To  kno.v  the  value  of  anj  number  of  figures. 
BULK. 

••.merale  from  the  rijjht  to  the  left   hand,  each  fig. 
•n  its  proper  place,  by  saying,  units,  tens,  hundreds, 
'MI mcration  Table. 

U-  value  of  each  figure,  join  the  name  of 
ihc  left  hand,  and  reading  to  the 
right. 

EXAMPLES. 

.'  the  folluicing  numbers. 

• 

Hundred  and  sixtv-on*. 
-four. 
;y-fifuv  thousand  and  twentjr-»ix\ 


MMl'LE    ADDITION.  17 

4G1,  One  hundred  and  twenty -three  thousand  four 
hundred  and  sixty -one. 

;MO.  Four  millions,  six  hundred  and  sixty-six  thou- 
sand two  hundred  and  forty. 

:.- — For  convenience  in  reading  large  numbers, 

<mv  be  divided  into  periods  of  three  figures  each, 
>  M    - 
as  follows  : 

987,  Nine  hundred  and  eighty-seven. 

.  Nine  hundred  and  eighty-seven  thousand. 
%7  000  ooo,  Nine  hundred  and  eighty-seven  million. 
987  634  3-M.  Nine  hundred   and   ei»hty-sr-ven   million, 

Fsix   hn tulied    and    fifty-four   thousand, 
three  hundred  and  twenty -one. 

To  u-rite  numbers. 

RfJtE. 

n  on  the  ri^ht  hand,  write  units  in  t!i?  uri's  place, 
tens  i.i  the  tens  place,  hundreds  in  the  hundreds  place, 
.,  towards  the  left  hand,  writing  each  figure  ac- 
conlin^to  its  proper  value  in  numeration;  tal: 
to  sui-piy  those  places  of  the  natural  order  with  cyphers 
v.hich  are  omitted  in  the  question. 

F.S. 

Write  down  in  proper  figures  the  fallowing  numbers 
Thirty-six. 

Two  hundred  and  seventy -nine. 
Thirty -seven  thousand,  five  hundred  and  fourteen. 
Nine  millions,  seventy-two  thousn: 
Eight  hundred  millions,  forty-  i^iud  and  iii'tv- 

!i\v. 


SIMPLE  ADDIT! 

J.S  putting  together   several   smaller  .of  the 

aame  denomination,  into  one  la:^,1--.  equal  to  the   whole 
or  sum  total  ;  as  4  dollars  and  six  dollars  ii.  is  10 

collars. 


]3  SIMPLE 

RULE. 

I  la  vino;  placed  umts  under  units,  tens  undtr-tens,  See, 

a  line  underneath,  and  be.uiu  with  the  units;  after 

Adding  MI  j  ire  in  tliat  column,  consider  how  ma- 

:.t:iinedin  their  sura:  set  down  the  remain - 

.  the  units.  a!i.x  tarry  so  many  as  you  have  tens, 

lunin  of  tons;  proceed  in  the  same  manner 

t  olrniin,  or  row,  and  set  down  the  \vhol« 


•!  PI.KS. 

(2.) 

(3.) 

"= 

^  ^ 

~  I 
-  .-     —  r-  — 

r^  J.^'  = 

r-  —  —  — 

1  \  -J 

1756 

<  1 

0452 

7  8 

•  2 

G  »i 

698 

7  4 

(11 

64179 

£  5  ; 

7  1 

.»  1 

71432 

45? 

3  2  : 

5  5  2  G  2  1 
546977 
4  1  3  r, 
521012 
876545 


(7.) 

S  7  1  4  5 
51714 
60845 
57857 
61784 
52501 


ADDITION.  19 


(28'i 

(9.) 

(too 

6  4 

5 

S 

4  1  2  S 

5 

2  6 

3  f 

1  7 

8  4 

5 

9 

3714 

£ 

7  1 

9  6 

5  7 

2  5 

6 

3 

7147 

3 

8  4 

I  9 

2  5 

4  1 

7 

i 

8321 

5 

3  1 

9  2 

5  1 

.3  8 

7  2 
4  1 

S 

9 

7 
5 

1457 
1726 

6 

3 

1  0 

7  1 

8i 

9  o 

7  2 

8  4 

3 

7 

2513 

2 

9  1 

4  7 

,(1 

) 

\ 

(12.) 

9  4 

2 

S 

1 

7 

8  2 

9 

571845 

6 

8  7 

7  4 

2 

1 

0 

6 

1  0 

8 

511704 

2 

2  9 

ti  1 

0 

0 

4 

o 

7  9 

6 

19460 

3 

7  2 

7  6 

o 

S 

1 

4 

5  7 

2 

8340 

7 

3  4 

6  0 

0 

0 

4 

1 

2  3 

4 

270 

1 

5  5 

7  0 

4 

1 

5 

6 

0  5 

3 

S  6 

0 

2  3 

5  6 

j* 

8 

0 

9 

3  8 

T 

1 

9 

5  0 

0  0 

5  G  2*4 

(1 
3 

3.) 
0  6 

4 

6 

(14.) 

£ 

5  9 

4 

6  2 

8 

1 

4 

5 

1 

S 

4 

0  0 

4  5 

2  1 

G 

0 

4 

3 

o 

5  4 

0 

4  4 

3  3 

8 

7  6 

1 

0 

4 

0 

5 

370 

5 

5  3 

2  6 

3 

4 

6 

o 

I 

4 

4  0 

5 

2  1 

7  4 

4 

0 

3 

0 

9 

4  0  6  4 

7 

6  2 

6  9 

9 

8 

2 

J^ 

2  0 

6 

8  5 

9  J 

|i t?*To  prove  Addition,  i/egin  at  the  top  of  the  sum, 
and  reckon  the  figures  downwards  in  the  same  manner  as 
they  were  added  upwards,  and  if  it  be  right,  this  .sum  total 
•will  be  equal  to  the  first:  Or  cut  off  the  upper  lino  of 
figures,  and  find  the  amount  of  the  rest  j  then  if  the/u  mount 
unper'Iine,  when  added,  be  equal  to  the  total,  the 
is  fctipposcd  to  be  right. 


SlMl'I.K    ADDITION. 

i  here  is  another  method  of  proof,  as  follows  : — 
Roject  or  cast  out  the  ninef;  in  each      EXAMPLE. 
nm-  or  sum  of  figure*,  and  set  down  the     3782     .«  2 
u-H i  directly  even  with  the     5766    °*  6 
its  row:  find  the  sum  (if  these     8755 

:    'icn  if  the  !  nines   * 

in  the  $'im  found  as  before,  is  IMJIKI!  to  the   18503 

i.iiu-s  in  t'u-  «u  m  total,  the  work      

.!  to  be  ri^ht. 

.  7421,   5063,   2196,   and    1245 
•ior  Jlns.  £6754. 

16.  Find  the  sum  of  3482,  783G45,  318,  7530,  and 
•45.  Jus.  10473020. 

17.  Find  the  sum  total  of  604,  4680,98,  64,  and  54. 

»#NS.  Fit? y-tivc  hundred. 
'A'hat  is  the  sum  total   of  24674,  16742,  34G78, 

.'//:<•.  One  hundred  thousand. 
Vdd  1021,  3489,  28765,  289,  and  6438  together. 
J/is.  Forty  thousand. 

•  is  the  sum  total  of  the  following  numbers,  viz* 
JO,  and 4005  ?  Jns.  11  111. 

>utn  total  ofthefollowmgnumbei 
(ircd  and  forty-seven, 
•j sand  six  hundred  and  five, 
thousand  six  humlm!. 
0  liunilred  and  c.levcn  t!ioi)s,and, 
ami  twenty-;' 

.  and  nine  thousan 


Jlnswtr,  Gl 374 177 


the  sum  of  the  following  numbers,  viz. 

••(1  and  «i- 

•  'ininlird  and  five, 

-ix  hundred, 

mdrrd  and  eleven  tluiusand, 
.iiid  twenty  -six 


9999999 


FEDEUAL    MO.VKV.  21 

QUKSTIONS. 

1.  "What  member  of  dollars  are  in  six  bags,  containing 
each  375-42  dollars  ?  An*.  22; 

2.  If  one  quarter  of  a  shin's  cargo  be  north  eleven 
thousand  and  ninety -nine  dollars,  how  many  dollars  is  the 
whole  cargo  worth  ? 

Jins.  44396  dols. 

3.  Moacy  was  first  mcde  of  gold  and  silver  at  Argos, 
eight  hundred  arid  ninety -f$<'-  ;--tme  Christ;  ho\v 
long  has  money  been  in  use  at  this  date,  1314? 

.  2708  year?. 

4.  The  distance  from  Portland  in  the    Province  of 
Maine,  to  Boston,  is  125  miles;  from  Boston  to  !••'• 
Haven,  162  miles;  from  thence  to  Neu  -York,  88  ;  from 
thence  to  Philadelphia,  95;  from  thence  to  Baltimi 
102:  from  thence  to  Charleston,  South-Carolina,  716; 
and   from  thence  to  Savannah,  119  miles — What  is  the 
whole  distance  from  Portland  to  Savannah  ? 

Jns.  1407  miles. 

5.  John,  Thomas,  and  Harry,   after   counting   their 
prize  money,  John  had  one  thousand  three  hundred  and 
seventy-five  dollars  :  Thomas  had  just  three  times  as  ma- 
ny as  John  :  and  Harry  had  just  as  many  as  John  aud 
Thomas  both — Pi-ay  how  many  dollars  had  Harry  ? 

Ans.  5500  dollars. 

FEDERAL  MONEY. 

JN  EXT  in  point  of  simplicity,  and  the  nearest  allied  to 
whole  numbers,  is  the  coin  of  the  I'nited  States,  or 

FEDERAL  MONEY. 

This  is  the  most  simple  and  easy  of  all  monev — it  in- 
creases in  a  tenfold  proportion,  like  whole  numbers. 
10  mills,  (m.)  make      1  cent,  marked    c. 
10  cents,  1  dime,  d. 

10  <!'  1  dollar,.  g. 

10  (lull,-,;-.  1  Easrle, 

Dollar  is  the  money  unit:  al(  other  ^(yUf^HB 
ing  valued  according   to  their   pla 
place.    A  point  or  comma,  called  a  .-•"/j/fl^s.r. 
placed  after  the  dollars  to  separate  them  from  t 


ADDITION   OF   TFDERAI.    MONET. 

denominations ;  then  the  first  figure  at  the  right  of  thii 
separatnx  is  dimes,  the  second  figure  cents,  and  the  third 
mills.* 


ADDITION  OF  FEDERAL  MONEY. 
RULE. 

1.  Place  the  numbers  according  to  their  value  ;  that  is, 
dollars  under  dollars,  dimes  under  dimes,  cents  under 
cents,  Sec.  and  proceed  exactly  as  in  whole  numbers  ; 
then  place  the  separatiix  in  the  sum  total,  directly  under 
the  separating  points  above. 

KXAMPLES. 

g.      d.c.m.  g.  d.  c.m.  g.  d.c.nti 

365.     541  459,  304  136,  5  1    4 

487,     060  416,  590  125,  090 

94,    670  168,  934  200,  909 

439,     089  239,  060  304,  006 

.500  143,  005  111,  1  9   1 


52128,     860,  j 

2.  When  accounts  are  kept  in  dollars  and  cent-.,  and 
no  other  denominations  arc  mentioned,  which  is  the  usu- 
al mode  in  common  reckoning,  then  the  two  first  figures 
kt  the   right  of  the  sej>aratri\  or  pt>inr,  may  be  called  so 
Many  rents  instead  ot  dimes  and  cents;  for  the  purr  uf 
"idv  the.  ten's  place  in  cents  ;  because  ten  rents 
make  a  dime;  for  example,  48,  75*  forty-eight  dollars, 
:i  climes  live  cents,  may  be  read  forty-eight  dollars 
•:its. 

dti^rrvod   tint  all  tho  figures  at  the  left  hand 
ix  nn-  dollars  ;  or  you  may  c;ill  the  first  figure 
"•r  t\-iu;!f*.  kc.     Tims  nny  sum   of  this 
differeothr.  either  wholly  in  tl 
ly  in  the  higher,  and  partly  in  thr  low- 
S7  Jit,  may  !)«•  *-ithor  rrnd  5>7.r>4  ecu' 

'  lollars  5  dimes  and  4  cents,  or 

;!c«.  7  (lollari  rj  dimes  and  1  co.nta. 


A.DDIT16N   Of   FEDERAL 


If  the  eeuts  are  leas  than  ten,  place  a  cypher  in  (he 
ten's  place,  or  place  of  dimes.  —  Example.  Writa  down 
four  dollars  and  7  cents.  Thus,  g4,  07  ots. 


EXAMPLES. 


1.  Find  the  sum  of  304 dollars,  39  cents;  291  dollars, 
9  cents;  136  dollars,  99  cents;  12 dollars  and  10 cents, 
39 


T304, 
I  291, 
I  136, 


,«.  j  «n,  09 

Thu3»      '}  13(£  99 

L  12,  10 

Sum,    744,  57    Seven  hundred  forty-four  doi 
—      lars  and  fifty-seven  cants. 

g.  cts,  g.  ct»« 

S64,  00  3287,  80 

21,  50  1729,  19 

8,  09  4249,  99 

0,  99  140,  01 


(6.) 
g.     cts. 

124,  50 
9,  07 
0,  60 

231.     01 

0,     75 

24,     00 

9,     44 

0,    95 


S  Vv'hat  is  tneaum  total  of  127  dols.  19  cents,  £78 
<Lk  19  cents,  34  dols.  7  ceuts,  5  dols.  10  cents,  and  1 
<*<>  Sj  cwnts  ?  j?ns.  S446,  54  ct-\ 


S4  ADDITION    OF    FEDERAL    MONET. 

9.  What  is  the  sum  df  378  dols.  1  ct.  156  dels.  91  cts. 
844  dols.  8  cts.  and  365  dols.  ?  Jus.  £1224. 

10.  What  is  the  sum  of  46  cents,  52  cents,  92  cents 
and  10  <*!  .*N«.  g2. 

11.  What  is  the  sum  of  9 dimes,  8 dimes,  and  80  cents? 

JHS.  82  A. 

12.  I  received    of  A   B  and  C  a  sum  of  money  ;   A 
paid  me  9  >  dols.  43  cts.    B  paid  me  justf  three  tint- 
much  as  A,  and  C  paid  me  just  as  much  us  A  and   li 
both  j  can  you  tell  me  how  much  money  C  paid  me  r 

Jinn.  $381,  72  cents. 

IS.  There  is  an  excellent  well  built  ship  just  returned 
from  the  Indies.  The  ship  only  is  valued  at  12145  dols. 
86  cents;  and  one  quarter  of  her  carjjo  is  worth  25411 
dols.  65  cents.  Pray  what  is  the  value  of  the  whole  ship 
and  cargo?  J/<s.  SI  13792.  46  cts. 


A  TAILOR'S  BILL 

•Vr.  Jamfs  Pay  well, 

T'>   Timothy  Taylor,  Dr. 
1814.  S. 

April  15.      To0-  yds.  ofClnth,  lit   ti,  30  p, 

oat, 

To  1 

To  £ 

in, 


I',;,  MI   n   t   ol  Conges*,  all  the  acr<-' 


SIMPLE   SUBTRACTION.  2S 

SIMPLE   SUBTRACTION.] 

Subtraction  of  whole  Numbers, 

1  EACHETH  to  take  a  less  nu^er  from  a  greater,  »f 
the  same  denomination,  and  thereby  shows  the  difference, 
or  remainder  :  as  4  dollars  subtracted  from  6  dollars,  the 
remainder  is  two  dollars. 

RULE. 

Place  the  least  number  under  the  greatest,  so  that  units 
may  stand  under  units,  tens  under  tens,  &c.  and  draw  a 
line  under  them. 

2.  Begin  at  the  right  hand,  and  take  each  figure  in  the 
lower  lire  from  the  figure  above  it,  and  set  down  the  re- 
mainder. 

3.  If  the  lower  figure  is  greater  than  that  above  it, 
add  ten  to  the  upper  figure  ;  from  which  number  so  in- 
creased, take  the  lower  and  set  down  the  remainder,  car- 
rying one  to  the  next  lower  number,  with  which  proceed 
a*  before,  and  so  on  till  the  whole  is  finished. 

1'iiooF.  Add  the  remainder  to  the  least  number,  and 
if  the  sum  be  equal  to  the  greatest,  the  work  is  right 

EXAMPLES. 

(I.)  (2.)  (3.) 

Greatest  nwnter,2  468      62157      879647:* 
L-ast  number,       1346      12148       1643489 

Difference, 


(4.)         (5.)         (6.) 
»  41G78S39    918764520     €5452167890 
Take  S154C999     91213806     12345697098 


SIMPLE   SUBTRACTION. 

(7.)  f8.)  • 

From  917144043G05      3562176255002 
Take   40600832164      1235271082165 


Rem. 

(9.)  (10.^          (11.) 

From    100000         2521 OT5         200000 
Take      65321         2000000          99999 

13.  From  560418,  take  29r>752.  3ns.  66666. 

14.  From  765410,  take  347-47.  dns.  730663. 

15.  From  341209,  take  198765.  3ns.  142444. 

16.  From  100046,  take  10009.  3ns.  90037. 

1 7.  From  2637804,  take  2576982.         3ns.  260822. 

18.  From  ninety  thousand,  five  hundred  and  forty -six, 
take  forty-two  thousand,  one  hundred  and  nine. 

3ns.  48437. 

19.  From  fifty -four  thousand  and  twenty-six,  take  nine 
thousand  two  hundred  and  fifty -four.          .'Lis.  44772. 

•CO.  From  one  million,  take  nine  hundred  and  ninety- 
Trine  thousand.  3n$.  One  thousand. 

51.  From    nine  hundred   and  eighty-seven   millions, 
*ake  nine  hundred  ami  eighty-seven  thousand. 

3ns.  986013000. 

.12.  Subtract  one  from  a  million,  and  shew  the  remain- 

,'JHs.  9P9999. 


QUESTIONS. 

1.  How  much  is  six  hundred  and  sixty-seven,  £reati:r 

•  hum] red  and  ninety-five  ? 
\  What  is  the  difference  between  twice  tu 

tiiree,  times  fortv -five  ?  ./MS.  HI. 

5,   llov.  much  r  than  565*iil  72[  iuljeit 

1  14. 

01:1  New-T.ondon  to    PKIladelpKia  is  C:o  miles. 

-London 

ward?  1  ia,  at  the    rate  of  .'>!)  miK-s  vach  day, 

u- would  '  fr«m  Philadelphia. 

45  miles. 


SUBTRACTION   Ol?    FEDKUAL    MONEY.  27 

5.  What  other  number  with  these  four,  viz.  21, 52, 16, 
and  12,  will  make  100  ?  wins.  19. 

6.  A  wine  merchant  bought  721   pipes  of  \vine   for 
90846  dollars,  and  sold  543  pipes  thereof  for  89049  dol- 
lars;  how  many  pipes  has  he   remaining  or  unsold,  and 
what  do  they  stand  him  in  ? 

.  178  pipes  unsold,  and  they  stand  him  in  g!7°~. 


SUBTRACTION  OF  FEDERAL  MONEY. 

RULE. 

Place  the  numbers  according  to  their  value ;  that  is. 
dollars  under  dollars,  dimes  under  dimes,  cents  under 
cents,  &c.  and  subtract  as  in  whole  numbers. 

EXAMPLES 

g.  d.  c.  TII. 
From  45,  475 
Take  43,  485 


Rem.  gl,    990  one  dollar, nine  dimes,  and  nine  cent*. 
or  one  dollar  and  ninety-nine  cents. 
g.     d.  c.          g.     d.  c.  TII.          g.     of.  c.  m. 
From    45,    74          46,     2  4  6        211,     1   1  0 
Take    13,     89          S6,     1  6  4         111,     114 


g.  g.    cts.  g.    eta. 

From   4284  411,  24  960,   00? 

Take    1993  16,  09  136,  41 

Rem.    '  ~_  " 

g.     cts.  g.    cts.  g.  cts. 

From   4106,71  1901,08  365,00 

Take     221,  69  864,  09  109,  01 

4l    11.  From  125  dollars,  take  9  dollars  9  cents. 

Ans.  SH5,  9 lets. 

•.*.  From  127  dollars  1  cent,  take  41  dollars  10 cents. 

Jlns.  g85,  91  cts. 


IM.K    Ml.' I.  1  1PL1CATJON. 

dollars  90  cents,  take  10: 

Jns.  i  i  is. 

1  1.  From  £49  dollars  43  cents,  take  18!)  dollar-. 

15.  From  TOO  dollars,  take  45  cts. 
I-'rom  ninety  dollars  and  ten  i 

eat*. 

17.  From  forty-one  dollars  eight  i  • 
nine  c  J/   - 

rroin  3  dols.  t:ik«.-  7  rts. 
19.  From  ninetv-nine  dollars,  take  ninetv-nine  cents. 

Jthts.  S98.  l  ct. 
30.  From  twenty  do!?,  take  twenty  1 1  i:N  and  one  mill 

J»/s.  8*10,  7<)rts.  0  mil  Is. 

21.  From  three  dollars,  take  one  hundred  and  ninety- 
nine  (  J/2S.  gl,  1  ct. 

From  20  dols.  take  1  dime.      J»is.  §10,  90  ct?. 

iiine  dollars  and  ninety  cen*s,  take  niuet.v- 
nincd.  J«s.  0  remains. 

'-  pri /.e  money  was  -219  dollars,  and  Tiioma* 
mucli,   lacking   -45  cents.     How 
!  Thomas  n-ceiyi-  :    Jjis.  Sy4;3T.  55  cts. 

'  I  pri/.c  money  to  the  amount  of 
lays  out  -111  dols.  41  i 

'.  dollars  ; 

i  a  suit  of  new  clothes:  besides  350  dol~. 
s    gambling.     How   much   will  ]\v. 
:'.0»-r  pa>iiii  !>i.-.  landlord's  bill,  which  .inm 
to  85  dols.  and  H  .  JJns.  g20,  58 


M  1'  1 .  i:    M  U  L  T I  P  L 1 C  A  T I O  N, 

E  TH  to  increase,  or  repeat  the  greater  of  iuo 
ten  as  there  are  units  in  the  I 
:  hence  it  performs  the  work  o§ma- 
'  compendious  manner. 

!»•  inulti])!  i 
.is  called  the  mulii 
number  fouiu;  -:i,  is  call-- 


SIMPLE    MULTIPLICATION. 


NOTE.  Both  multiplier  and  multiplicand  are  in  gene- 
ral called  factors,  or  terms. 

CASE   I. 
When  the  multiplier  is  not  more  than  twelve. 

RULE. 

Multiply  each  figure  in  the  multiplicand  by  the  multi- 
plier; carry  one  for  every  ten,  (as  in  addition  of  whole 
numbers)  and  you  will  have  the  product  or  answer. 

PROOF.* 
Multiply  the  multiplier  by  the  multiplicand. 

EXAMPLES. 

"What  number  is  equal  to  3  times  365  ? 

Thus,    365  multiplicand. 
3  multiplier. 


Multiplicand    74635 
Multiplier  5 


s.  1095  product. 
5432  2345  9075 

456 


Product 


71034 


1432046 
11 


£240613 
12 


4684114 
12 


CASE  II. 

When  the  multiplier  consist*  of  several  figures. 

RULE. 

The  multiplier  being  placed  under  the  multiplicand 
units  under  units,  tens  under  tens,  &c.  multiply  by  i 
significant  figure  in  the  multiplier  separately,  placing 
first  figure  in  each  product  exactly  under  its  multiplier; 

*  Multiplication  may  also  he  proved  by  casting  out  the  9's 
in  th.>   two  factors,  and   setting  down  the  remainders ;  then 
multiplying  the  two   remainders  together  ;  if  the  excess  of 
8V  in  then  product  is  equal  to  the  ^excess  oi 
i,  the  work  is  supposed  to  be  right. 
3* 


30  f  SIMPLE   MULTIPLICATION. 

then  add  tin;  several  products  togethernn  *Hc  same  ordin 
as  they  stand,  and  their  sum  will  be  the  total  product. 

EXAMPLES. 

\Vhat  number  is  equal  to  47  times  365  ? 

Multiplicand     365 
Multiplier  4  7 

2555 
1460 


Ans.  17155  product. 

Multiplicand,  37864  34293     47042 

Multiplier,     209  74       91 

340776 

75728 

Product,    7913576  2537682    4280822 

8253       25203  2193       9876 

826        4025  4072       9405 


6816978    101442075     8929896    92883780 


269181        261986         40634 
4629         7638        42068 


1246038840     2001049068     1709391  IS-? 


134092          918^ 
87362  100 


1171 


. 
.iat  is  1he  loUl  product  of  7 

i. at  number  is  ctjual  to  4(," 
.in*. 


SIMPLE    MULTIPLICATION.  SI 

CASE  III. 

'When  thei'2  are  cyphers  on  the  right  hand  of  cither  or 
both  of  the  factors,  neglect  those  cyphers;  then  place  the 
significant  figures  under  one  another,  and  multiply  by 
them  only,  and  to  the  right  hand  of  the  product,  place  as 
many  cyphers  as  were  omitted  in  both  the  factors. 

EXAMl'tBS. 

21200  51800  §4600 

70  36  54000 


1484000  1144800          2876400000 


55926000  82530     . 

5040  98£60000 


I092150400CO  8109397800000 


7065000x8700=61465500000 

7496431300x695000=521001885000000 

360000X  1200000=452000000000 

CASE  IV. 

When  the  multiplier  is  a  composite  number,  lhat '«, 
when  it  is  produced  by  multiplying  any  two  numocrs  m 
the  table  together ;  multiply  first  by  one  of  those  figures 
and  that  product  by  the  other;  and  the  last  product  WJH 
be  the  total  required. 

EXAMPLES. 

Multiply  41564  by  55. 
S5.  7 

289548  Product  of  7 
5 

1447740  Product  of  35 

2.  Multiply  764151  by  48.  Jlns.  56678288. 

3.  Multiply  542516  by  56.  .ins.  1018089G. 

4.  Multiply  209402  by  72.  .4ws.  lo07GJ>44. 
•  .  Multiply  91738     by  81.  *lns.  7430778. 
V).  Multiply  34462    by  108.  .Ins.  5721896. 
7.  Multiply  615243  by  144.  Anf.  885S4992. 


$5  SIMPLE   MULTZFLICATIOW. 

CASE  V. 

To  multiply  by  10,  100,  1000,  &c.  annox  to  the  mul- 
tiplicand all  the  cyphers  in  the  multiplier,  and  it  will 
make  the  product  required. 

EXAMPLES. 

1.  Multiply  365    by  10.  Ans.  3650 

2.  Multiply  4657  by  100.  jJns.  465700 

3.  Multiply  5224  by  1000.  '  ^flns.1  5224000 
4    Multiply  26460  by  10000.             ^K.«.  264600000 

EXAMPLES    FOR   EXERCISK. 

1.  Multiply  1203450  by  9004.  ^JiS.  10835863800 

2.  Multiply  9087061  by  56708.  Jins.  515309055188 

5.  Multiply  8706544  by  67089.  Jliis.  584113530416 
1.  Multiply  4321209  by  123409.  Ms.  -533276081481 
j.  Multiply  3456789  by  5C7090.  Jus.  1960310474010 

6.  Multiply  8496427  by  874359.  Jins.  74289274] 5293 

98763542x98763542=9754257223385764 

Application  and  Use  of  Mult iplicailon. 
In  making  out  bills  of  parcels,  and  in  finding  the  value 
of  goods  ;  when  the  price  of  one  yard,  pound,  Sec.  is  giv- 
en (in  Federal  Money)  to  find  the  value  of  the  whole 
quantity. 

RULE. 

Multiply  the  given  price  and  quantity  together,  as  iu 

whole  numbers,  and  the  separatrix  will  be  as  many  figures 

the  right  hand  in  the  product,  as  in  the  given  price. 

EXAMPLES. 

1.  What  will  35  yards  of  broad-  >  g.  d.  c.  m. 
cloth  come  to,  at  S  3,  4  9  6  per  yard  ? 

5  5 


Ans.  SI 22,  .3  6  08=122  doi- 

[lars,  36  cents. 

What  cost  55  Ib.  cheese  at  8  cents  per  Ib.  ? 
,08 

.  22,  80«=2  dollars,  80 


SIMFtE    MLI.TIPLICAT10N.  58" 

hat  is  the  value  of  29  pairs  of  men's  shoos,  at  1 
51  cents  perpa.il1?  Ms.  &43,  79  cents. 

4.  What  cost  151  yards  of  Irish  linen,  at  38  cents  per 
yisi-J  ?  Ms.  £49,  78  cents. 

5.  What  cost  140  reams  of  paper,  at  2  dollars  35  cents 
per  ream  ?  Ms.  g329. 

6.  What  cost  144  Ib.  of  hyson  tea,  af  3  dollars  51  cents 
per  Ib.  ?  Ms.  8505,  44  cents. 

7.  What  cost  94  bushels  of   oats,  at  33  cents  per  bush,; 
el  ?  Ms.  £31,  3  cents. 

8.  What  do  50  firkins  of  butter  come  to,  at  7  dollars 
14  cents  per  firkin  t  Ms.  §357. 

9.  What  cost  12  c\vt.  of  Malaga  raisins,  at  7  dollars 
31  cents  per  c\vt.  ?  Ms.  SS7,  73  cents. 

10.  Bought  57  horses  for  '.hipping,  at  52  dollars  pot 
head  ;  what  do  they  conn  Ms.  S1924. 

11.  What  is  the  amount  of  500  !bs.  of  hog's-lard,  at  15 
cents  per  Ib.  r  Ms.  g75. 

12.  What  is  the  value  of  75  yards  of  satin,  at  3  dollars 
75  cents  per  yard?  Mt>.  8281,  25  cents. 

13.  What  cost  367  acres  of  land,  at  14  dols.  67  cents 
per  acre  ?  Ms.  S5385,  89  cents. 

14.  What  does  857  bis.  pork  come  to,  at  18  dols.  9S 
cents  per  bl.  ?  Jns.  SI  0223,  1  cent 

15.  What  does  15  tons  of  Hay  come  to,  at  20  dols.  78 
cts.  per  ton  ?  *  Ms.  S3  11,  70  cents. 

1G.  Find  the  amount  of  the  following 
BILL  OF  PARCELS 

New-London,  Marcn  9,  1814. 
Mr.  James  Paywell,  Bought  of  William  Merckaxt, 

S.  cts. 

28  Ib.  of  Green  Toa,  at  2,  15  per  Ib. 

41  Ib.  of  Coffee,  at  0,  21 

54  !b.  of  Loaf  Sugar,  at  0,  19 

13  cwt.  of  Malaga  Raisins,  at  7,  31  per  cwt. 
'35  firkins  of  Butter,  at  7,  14  per  fir. 

.•airs  of  worsted  Hose,   at  1,  04  per  pair. 
!U  bushels  of  Oats,  at  0,  33  per  bush.' 

"33  pairs  of  men's  Shoes,      at  1  ,  1  2j?pr  pair.       » 

Wm 


rcclved  payment  in  full,  WILMAM 


or  WHOLE  NUMBBII&. 


A  SHORT  RULE. 

NOTE.  The  value  of  lOOlbs.  of  any  article  will  be  just 
as  many  dollars  as  the  article  is  cents  a  pound. 

F«ir  'lOO  Ib.  at  1  cent  per  Ib.  =  100  cents  =  1  dollar. 

100  Ib.  of  beef  at  4  cents  a  Ib.  comes  to  400  cents«=4 
dollars,  Sec. 


DIVISION  OF  WHOLE  NUMBERS. 

SIMPLE  DIVISION  teaches  to  find  how  many  ttmes 
one  whol*  number  is  contained  in  another  ;  and  also 
what  remains;  and  is  a  concise  \vay  of  performing  seve- 
ral subtractions. 

Four  principal  parts  are  to  be  noticed  in  Division  : 

1.  The  Dividend,  or  number  <j;iven  to  be  divided. 

2.  The  Divisor,  or  number  given  to  divide  by. 

5.  The  (Quotient,  or  answer  to  the  question,  which 
shows  how  many  times  the  divisor  is  contained  in  the 
dividend. 

4.  The  Remainder,  which  is  always  less  than  the  di- 
visor, and  of  the  same  name  with  the  Dividend. 

RULE. 

.  seek  how  many  times  the  divisor  is  contained  in 
as  many  of  the  left  hand  figures  of  the  dividend  as  are 
jn^t  necessary,  (that  if)  find  the  greatest  figure  that  the 
divisor  can  be  multiplied  by,  bu  as  to  produce  a  product 
if  the  part  of  the  dhidend  used)  \\heu 
found,  pl.it  i'  the  figure  in  the  quotient  :  multiply  the  di- 
vj-ur  by  t;>i>  «|i:o.  ....-;  place  the  product  under 

that  part    of  the   dividend  used  ;  then  subtract  it  there- 
from, aiulbrin^  down  the  next  figure  of  the  dividend  to 
Kind  of  i  .der  ;  after  \vliich,  you  must 

:.iiply  and  subtract,  till  you  ha\e  brought  down 

ihe  dividend. 

Multiply  the  divisor  and  quotient  together 

nnd  add  the  remainder  if  thn-e  be  any  to  t!ie  pryduo4- ;  if 

mi  will  be  equal  to  the  divide;, id.* 

*  Anot;  '1  \\hich  some  r.o  prove  divi 

li.in  is  ;i.->  -'i?,.  Add  '!  <icr  .tnd  all   the  pro- 

ducts of  the  several  quotient  figures  multiplied  uy  the  diviaor 


DIVISION   OF   -\VHOLE    NUMBERS. 


35 


EXAMPLES. 

1.  I  low  many  times  is  4  2.  Divide  3656  dollar* 

contained  in  9391  ? 


Divisor+Div.  Quotient. 
4)9391(    234r 
8  4 


equally  among  8  men. 
Divisor, Div.  Quotient. 
8)3656(457 
32 


13          9388 

45 

12            +3  Rein. 

40 

19       9391  Proof. 
1$ 

31 

98 

56 
56 

3656    Proof  by 
addition. 

3  Remainder. 

Divisor,  Div.  Quotient. 
29)15359(529 
145 

365)49640(156 
365 

Proof  by       
excess  of  9's        85 
5                       58 

1314 
1095 

2X7            m 

2190 

5                        261 

2190 

Remains      18 

0  Rem. 

together,  according  to  the  order  in  which  they  stand  in  the 
work  ;*and  this  sum,  when  the  work  is  right  will  be  equal  to 
the  dividend. 
A  third  method  of  proof  by  excess  of  nines  is  as  follows,  viz. 

1.  Cast  the  nines  out  of  the  divisor  and  place  the  excess 
on  the  left  hand. 

2.  Do  the  same  with  the  quotient  and  place  it  on  flie  right 
hand. 

8.  Multiply  these  two  figures  together, and  add  their  pro- 
duct to  the  remainder,  and  reject  the  nines  and  place  the  ex- 
cess at  top. 

4.  Cast  the  nines  out  of  the  dividend  and  place  the  excess 
at  bottom. 

NOTE.  If  tbp  win  is  right,  th«  top  and  bottom  figures  witt 
#e  uffi&e. 


DIVISION  OF  WHOLE  NUHBEUS. 

J)ivisor.I)iv.(luotient.  95)85595(901 

61)28609(469  7.16)863" 

-r  5)251  l')-K  there  remains  664 

9.  Divide  189:3:312  by  912.  3ns.  2076. 

10.  Divide  1H-U312  by  2076.  Jlns.  9 1 

11.  Divide  472.54149  by  467-4.        dns.  10110, 

.  >Vhat  is  the  quotient  of  330098048   divided  bj 
4-207  ?  .'!??.«.  78464. 

13.  What   ts  the  quotient   of  701Sj8463  ilivided  bv 
8465  ? 

14.  How  often  does  7618584G5  contain  90001  ? 

J/ 

15.  IIo\v  many  times  58473  can  you  have  in  1 19l84v' 

Ans.  SO1.- 

16.  Divide  £80208122081  by  912:314. 

quotient  307140. 

MORE    EXAMPLES    FOR    KXKUriSK. 

Divisor.  Dividend. 
>( 

4761 4)327879 1. S6(  )  9! 

987654)98864 1G54(  )  ...0 

CASE  II. 

When  there  nre  cyphers  at  tin*  right  liaiiil  r- 
v pliers  in  the  divisor,  and   ; 
ii-nm  tiie.  ri^ht  hand  o!' 
••  remaining  ones  ;is  usual,  a..d   to 
y";  annex  those  li^ui'es  tut  otf 
and  y»»u  will  liave  the  true  i  - 

EN 

1.   Divide  467:,6-.M  b . 

truequoti 


\7f\ 


COXTHACTIOXS   IV   DIVISION.  3* 


2.  Divide  57P43C6~fl  by  6500 

5.  Divide  4<2  i4i!0  00  hv  49000  ^fl*.  8600 

4.  Divide  11659112     by  890000.  Jin*.  1SUAVW 

5.  Divide  9187642      by  9170000.  Ans. 


MOtE    EXAMPLES. 

Divisnr.  Dividend.  Remains, 

125000)436250000fQ«ofie»t.)         0 
1  20000)  149596478(  )7'*47S 

90  1  000)  0543472  ,r;l  •(  V2£  1  530 

720000)987654000(  )534000 

CASK  III. 

Short  Division  is  when  the  divisor  docs  not  exceed  1£ 

RULE. 

Consider  how  many  times  the  divisor  is  contained  ill 
the  1ir*t  figure  or  iiiriires  of  the  dividend,  put  the  result 
under,  and  carry  as  many  tens  to  the  next  figure  as  ill  ere 
are  ones  over. 

Divide  every  figure  in  the  same  manner  till  the  whole 
i%  finished. 

EXAMPLES. 
Divisor.  Dh'idend. 

2)113415         3)85494        4)39407        5)94379 

Quotient  56707—1 
S)120G16  7)152715  8)96872  9)118724 


11)6986197  12)14814096  12)570196582 


t  Contractions  in  Division. 

"\Vhen  the  divisor  is  such  a  number,  that  any 
tires  in  the  Table,  beinj;  multiplied  together  will  produce 
ft.  divide  the  given  dividend  by  one  of  those  figures  ;  the 
fj!ii»ti«'nt  thence  arising  by  the  other  ;  and  ifiie  last  quo- 
tient will  be  the  answer. 

NOTE.    The  total  remainder  is  fotiml  !r  rsurtir)  ving 
ilie  last  remainder  bjthc  first  dtvigor,  and  j^ciog  »a  the 


SITPLEMENT    TO    MULTIPLICAT1OK. 

EXAMI'LKS. 

by  7-2. 

or  8)16:2641  last  rein.  T 


i  .''>.•?()—  I  X§ 

—7                                C-23  8—  8  GS 

'  —  .  —        fr*i  ran.  -f2 
Tru*  Quotient  xM- 


2.   Dh-itlp   irS4i"54     by   Hi. 

3.  n 

. 

6.  Pivi-le   I  l-l^ro     by   48. 

7.  J/i.s. 

8.  Divi.;  i  «£/, 

9.  I-  l»v   103. 

10.  i)ivi'.  i44.  J» 

2.  To  divide  by  10,  100,  1COO,  &c. 

MILE. 

Cat  of!'  as  ma;r  '  .if  tluxtivt- 

t  'S  ill  lli('  (I1-,  l-ui".  ;ii 

der  ;  and  the  other  figure*  oi'  tliC 

div:  i;t. 

\  Ml'I.ES. 

1.  Divide  by  '  36  and  5  rr 

2.  ' 
3. 


•.^KNT  TO  MULTU'LH'AT! 


To  mn'  f«  •-,!>.-  I; 
jn. 


.  ati;l  tukc  i,  i   j  &. 


SUPPLEMENT  TO  MULTIPLICATION*. 

EXAMPLES. 

Multiply  57  by23i.  Multiply  48  by  2$. 

2)37  48 

23$  .    22 

135  24  =  £ 

111  J'=i 

74  9t> 


j  Jinvtcer.  .?«*.  ^ 

S.  Multiply     £11   by  50Jv 

4.  Multij.i'v  '-2404  by     8£. 

5.  Multiply     34J  by  wSws.     ( 

6.  Multiply  6497  by  wJ«s.  SS41  $ 

Questions  tr>  E.vcrt-isf  J/  tlti;ni  -ation  ami  J)ifi*ion. 

1.  \Vhat  will  1)4'  tous  of  hay  ct*Kie  to,  at  14  <!<•          R 
ton  ? 

2.  [fit  takes  320  cods  to  make  a  inile,  and  <•  ei 
•ontuins  5-i  yards;  ho\v  many  yar;Ua"<'  t!u  i-e  i  •: 

JMS.    17 

5.  Sold  a  ship  for  11316  dollars,  and  I  n 
\viiat  \'-  as  inv  part  of  tl;e  money  r 

4.  In  27G  !>aircis  uf  raisins,  each  3 i  cwt    !>;.•-.   iiiaoy 
hundred  \\oiv.htr  Ait*. 

o.  In   3I)  j.ieces   of  cloth,  each  piece   con; 
yards:  h-»\v  inary  yaids  in  the  \.hole?    dn>: 

6.  What  Is  the  product  "f  161  iiHiitipiietl  bv  ir 

7.  K*  a  man  spends  482  <l  ft  liars  a  year,  \vi:a»  is  ti  at  per 
oa'eudar  mu.i 

of  i".5   ir.en   tf.ok  .'i    pri /    . 
«<j;ia!iy  divi-A'd  amo:)^  i 
what  i>  the  val'ie  of  V\>>  !>ri/ 

9.  What,  uuii'ber  multiplied  Uy  9,  \\ill  mu 

J/«s.  25. 

10.  '!' e  •  •!  >iicnt  of  a  certain  number   >  -4;r.  a 
i-  ii!»"ul  :  .1 

11   \Vua-  ,  af  3.s.  pri  v:. 

IS.   Vi  in*  cost  45  oxen,  at3A  per  hem 


40  'COMPOUND  ADDITION. 

13.  What  cost  144  Ib.  of  Indigo,  at  2  dols.  50  cts.  or 
250  cents  per  Ib.  .0  »w.  gSGO. 

14.  Writedown  four  thousand  six  hundred  and  seven* 
teen,  multiply  it  by  twelve,  divide  the  product  by  nine, 
and  add  36j  to  the  quotient,  then  from  that  sum  subtract 
five  thousand  five  hundred  and  twenty-one,  aJid  tiie  re- 
mainder will  be  just  1000.    Try  it  and  see. 


COMPOUND   ADDITION* 

IS  the  adding  of  several  numbers  together,  having  dif- 
ferent denominations,  but  of  the  same  generic  kind,  as 
pounds,  shillings  and  pence,  &c.  Tons,  nuudreds,  quar- 
ters, &.C. 

RULE.» 

1.  Place  the  numbers  so  that  those  of  the  same  denom- 
ination may  stand  directly  under  each  other. 

2.  Add  the  first  column  or  denomination  together,  as 
in  vvh.i'e  m:  nber.-i :  tiien  divide  the  sum  by  as  many   of 
the  same  denomination  as  make  one  of  the  next  Crater; 
sottiij^  iK-.\  n  tin-  remainder  under  the  column  added. 

v  the    quotient  to   the  next  MI;M-  ior  denominp 
nuing  the  same  to  the  last,  which  add,  as  in  simple 
•addition. 

I.  STERLING  MONEY, 

Ji?  the  money  of  acrout't  in  Great-Britain,  and  is  reck- 
oned in  t'.i'.inds,  fcluiliu^,  Pence  and  R.  ^ee 
the  Police  Tables. 

*  T  •  'Mltion  of  tliis 

itir  J/IM«';'  i  -I  in    thr  failliin^s ;   1  in 

\li>-  -  •  l»'m    iht-  .'I    I  in  tlu-  pounds,    to  20 

in  the    »l.hii.;^s  ;   tin  !•    ar- 

<y    in 

in  thi:  addition  of  compound  u.jialers  of  aw 


COMPOUND    ADDITION.  41 


EXAMPLEb. 

"What  is  the  sum  total  of  \7L  13s.            f 
Qd.—  194.  2.s.    9id.—  14/.  10s.    11  id. 
aud  12/.  9s.  lid.?                              Thosj 

£ 

47 
19 

14 

,   s     d. 

15      6 

9     1J 

Jlnsi 

tvr,£. 

16 

4i 

17 

9, 
13 

11 

(5) 

£.     s.     a.   qr. 
h4     17     5     3 

/:.  s. 

SO     11 

d.  ar. 

13 

10 

2 

75      13 

4 

2 

15      10 

9     1 

10 

17 

5 

50     17 

8 

2 

1 

0 

1     2 

8 

8 

7 

20      10   10 

1 

S 

9 

8     S 

5 

S 

4 

16 

5 

0 

4 

(i 

r>    i 

^ 

P5 

(6) 

) 

4 

s. 

d. 

</r. 

£  .     s. 

il. 

qr. 

£.        s.     rf.  ?f 

47 

17 

6 

2 

~7     17 

10 

3 

541 

0     0 

0 

3 

9 

10 

3 

GO      6 

8 

0 

711 

9     8 

1 

59 

17 

11 

2 

7     14 

11 

2 

918 

6     9 

S 

317 

16 

9 

3 

18     19 

9 

3 

1  M) 

15   10 

1 

762 

19 

10 

1 

91      15 

8 

3 

soo 

19  11 

3 

407 

17 

6 

2 

18     17 

10 

3 

48 

10'    7     Z 

1 

19 

9 

5       0 

1 

2 

0 

1 

!     9 

S 

(S) 

(9)                           (10) 

£. 

s. 

d. 

£. 

s. 

a. 

£. 

s. 

d. 

1U5 

17 

6 

940 

to 

7 

11 

193 

10 

11 

36 

9 

11 

20 

0 

4 

901 

13 

0 

11 

4 

10 

1 

to 

519 

19 

7 

141 

10 

6 

17 

11 

9 

48 

17 

4 

126 

14 

0 

9 

1( 

104 

11 

9 

104 

19 

7 

0 

1! 

90 

16 

7 

too 

10 

6 

19 

J 

111 

g 

9 

100 

0 

0 

11 

10^ 

0 

10 

9 

0 

9 

1  oO 

14 

449 

12 

G 

0 

19 

6 

421 

10 

Si 

29 

10 

4 

120 

0 

8 

15 

4 

flOMPOVND    ADDITION. 


1  the  amount  of  the  following    £.     s.     d, 
-III.    H)s.  —  I/.  [ 

.  Oi.  r^.—  ito  4;J.—  2r;.r* 


£.  115     7 


•nil  fiJ&— -34f.  10s.  7c/.— :/.  18s.  5c*. 

«— c-tr.. 

15.  V  urn  total  of  14J.  19s.  6rfT—lL 

u.s.  Grf.~ -  -  '.and  Os. 

.  5d. 
1-1.  amount  of  the  followipg  sums.  viz. 

1  "•"•£•      s-     <^> 

-     - 

•  rids,    1111:  ,     - 

. ,    . .  -.-.---- 


,1      6 


!.•>.  How  much  is  the  sum  of 

;i!id  six  pence,    * 

.,;..... 

e  j>cncc,     -     - 

.      .  .:,     - 

. 


Jim.     . 


If/.  Bo^'Jit  aquarv'  '.'.  lO.s.  paid  for 

mty-aine 

'  u-u 

•-iuHiij^s  au»l 
i:at  iliii  l  and  jr.e  in  ? 

.   1;>(i  4s.  Id. 

•n  took  a  pri/.e,  and  i\n\  it  ecjnally 

ami.  ;  ?\vu  liundied  and  lorrv 

:   :-,«'\*-n  jH-iice  ;  how  rnucli 
. 

M-14  IS.  $$. 


00VPOUKB    ADDIYJAii,  43 

2.    TUOV  WEIGHT. 


tf. 

16 
4 
8 
6 

'    4 
0 

OK.   pict.  »r.              t'i.     os.   p'it»i.  <r;\ 
11      19     23                  8     11      19     21 
4     16     21                  6     10     16       8 
8     19     14                  7       8     17     21 
9     14     17                 468     23 
7     10      7                 9      7     14     17 
7     11     12                 7       9     13     10 

«rf.  gr.  Ib. 
2      3     27 
1       1      17 
4      2     26 
6      1      13 
^      S       15 
6      2      16 

3,    AVOlRnUPOIS     WKIGHT. 

flj.    oz.    dr.      T.    cu't.  (jr.   lh.    ox.  dr. 
24     13     14        91      17      2     24      13     14 
17     12     11         1(>       9      0      17      10     12 
£6     12     15        14      13      2     04        9     11 
16      8       7        47      11      3      19      14      5 
24     10     12        69     00      1     00      00     12 
11     12    12       77     19     3     27      15     11 

5  3  *rr. 

9      1       17 
5     2       0 
6     1     17 
4     0      16 
5     2     12 
6     1      10 

4.    Al'OTHRCAHIES    WEIGHT. 

5    5    ;•   £r-              fc    §    5  3   gr- 
id    7     2     19             12     11     6     1     15 
6     3     0     12               4970l;i 
7617              9     10    1     2     16 
9     5     2     12               4-812     iy 
6  .1     0     16               9       001 
9     S     2     19               49216 

yd.   qr.  nm. 
71     3     3 
IS     2     1 
10     0     I 

42     3     ft 
57     2     2 
49     2     3 

C.     CLOTU     MF.ASUttP^. 

*  JK.  E.  (jr.  na.                 IS.  F.  qr.  rut. 
44     3     2                              '21 
49     4     3                         07     1     6 
06    2    3                        76    0     G 

84    4    i                   ,v:    2    r, 

07     0     0                          53     2     e 
61     2     1                         09     2     ii 

44 


f.  pi 

1  7     1 

2  i 

1      5     0 
2 

261 
300 


COMPOUND    ADDITION, 
6.    OKY     MEASURE. 

bu.  pk.  t]t. 

17  2  5 

34  2  7 

33  S  G 

16  S  4 

27  2  6 

50  0  7 


71 


261 
0 


64 

4.1  0  4 

52  3  5     1 

94  2  3     0 

34  3  7     0 


gat.  ijt.  pt.  ?i. 
39    5     I 
17    2     1     2 


H     0     0 
4(1     2      I 


*     I 

2 


,  .    V  TXi:     MEASURE. 

hint.  gal.  tjt.  pt. 

42  (SI     3     1 

27  59     2    0 

9  14    0     1 
0       921- 

1G  G4     1     1 

4  00     3     0 


tun.klul.gnl.ijt. 


2  3-',  2 

1  59  I 
2 

0  52  2 

17      3  11  1 

i)      1  9  0 


54 
19 
23 
19 


/'•'.  in.  I.e. 


-i 

2 

II 

2 

3 

1 

8 

I 

1 

2 

9 

S 

6 

2 

10 

1 

1 

0 

6 

8 

3 

1 

7 

0 

8.    LOXli     MF.AfWRE. 


4ti% 
58 


16 


9    ( 
17     4      18 

7    5     i.; 


fr.  m.fur.  po. 
86  2  6  x  3G 
52  1  7 

i      19 

1  4      15 

2  3 


.  o 

7 


'• 


:r 


9.    I.AXD    Oil     SqVA: 

.rii 


19 
9 


1 


5 
6 
8 
0 
4 


143 
34 


10.   SOLID    MEASUKX. 

&'.   ft.     <tord*.   fed.    feet.  inclits.. 

41    43        $    *122       13  144S 

12    45        4    114       16  '  1?26 

49    6       7    83       S  866 

4    27       10*   127       14  284 


11.  TIME.  e 

Y.  m.  u.\  da.  1'r.  da.  Ji.  m.  sac, 

57  11  3  6  24  336  23  54  ^4 

3  9  2  S  21  40  12  40  £4 

29  8  2  5  IS  112  14  00  17 

46  10  2  4  14  9  11  18  14 

10  7  1  2  8  54  3  16  13 


MOTION. 

S.    •      '       •                  S.     *  '  • 

5  29     17     14                  11     29  59  59 
1       6     10     17                    0    00  40  10 
4     18     17     11                      94  10  49 

6  14     18     10                     4     11  6  10 


I  O  M  PO  U  N  D  S  U  JBT  R  AC  T  1  ON, 

1  EACHES  to  find  the  difierenr^;,  inequality  or  excew- 
bctwi:cii  any  two  sums  of  diverse  denouilaatiooa. 
RULE. 


Place  those  numbers  umlei-  o.rjcli  oihor,  which  ar«  of 
•WiesaUiC  thoicss!  .'<^rea<UjT5 

begin  >vit!i  t!i*j  le-i.->t  denomination,  and  ;!"  it  eitce- 
fiyinv  OVIT  it,  b"i-;-u\v  as  mauv  units  a#  m~'.  t'.e 

);t'^t  .;'  cater  :  siibtraet  it  therefrn  v:ij*'er*u»« 

a<!4t!ie  upper  fivute.rejne.rnben  i  >(3w 

nextcup«rior  rienouiinatiou  fertiat  wliisbyon  &«r:'fl 


4u  COMPOUND 

NOTE.     Tiie  iiietuuJ  of  proof  is  the  same  as  in  simple 
subtraction. 

EXAMPLES 

1.  £lfrling   •Mtniey. 

0)  '    (2) 

f..  .s.  d.  qr.  £.  s.  </.  qr. 

I'fTim  53  14  1  ; 

128  17  4  2  10  19  G  3 


17 


£.     >'   f/.  .  rf.  grr 

'    10  2  ,            J.ent  082 

11   8  Iteurivcd  IS  10  7  3   ' 

.ins  -^  • 

id        Due  to  me 

(C)  (*)  (B) 

£.     s.     rf.  £.  8.  J.  qr.  ,.   d.  qr. 

i       5     0     0  71112  476  10  9  1 

Take       4   10   11  4   17  S  1  277   17  7   1 


(D)                       (10)  (11) 

•    »•  (/r'        £•  y-  "•  £•  -*•  "•  9 

'18  ',  7   1 

1  1 ;   1 9  8  0963 


7l.  11s.  and   paid    l^/.    17;:.   fx,'. 

- 
13.   i 

14. 

v:usand  t;vo  liundi 

.  £7131 


COMPOUND   SLBTRACTIOV.  47 

16.  How  much  d-.'es  seven  hundied  ami  tlyht  pounds, 
exci'tM  tisirfv-nirie  pound?;,  til'ttreh  shil!'  -ti  pence 

JK.S. 

17.  From  one  hundred  pounds,  take  lour  miiic-* 
per 

.  18"  Received  of  four  inen,t  urns  of  money, 
vj/.   Tin*  first  paid  me  37/.    IK  - 

7</.  ti.<>  third  ID/.    !  s  mwch  .is  ;.H 

the  of'icr  t!  !--'0,  lacking  19«.  UtY.  I  d                        ;io'.e 

sum  received'?  wj?2i\   'ltij  5s.  4rf. 


53.   THOY  WRIGHT. 

Ib.  oz,  pet.   oz.  pu-t.  sr.  /'-•  o-r. 

From    (i  11   14    4  19  'sJl  44  9   «  ].- 

Take    2   5  1G    2  14  5.3  17  3  16  IS 


Rcm. 


bap.  pict.   s;r.         it),    os.  pivt.    %r 

2     10     14  942     2       0       "0 

G83     1       9     13  892     9       2         3 


3.  AVOIHOUPOISE    WF.IOHT. 

V).  oz.  dr.  C.  <jr.  l,lj.     T.  cwt.  qr.  l!>.  oz.  dr. 

7   9  12  7  3  13     7  10  3  IT  5  12 

3  12   9  5  1  15     5  13  1   19  10  9 


T.  cirt.  qr.  Ik.  oz.  dr.  T.  curt.  qr.  Ib.  o.z.  dr. 
610  11  0  20  II)  11  S17  12  1  12  9  12 
193  17  1  20  12  14  180  12  1  14  10  14 


4.  AVOTHECAIUES'    WEIG.H'I. 

ih     $    5  39   gr-            tb    5  5  D  gr. 

1'9        87  4     I       17                 55      7  3  1  14 

9116  1     £      15                17    10  6  1  18 


5.  TT.OTH 


43 


I"/.  (?'  I?./'.  <jr.  E.F".  rr 

I      5  407     :,     I 

19     1     3  "     °.  1-49     2      1 


IV.  gr.  na.  '.  </r.  .VT.  JR.  FL  qr.  net. 

SI. i     fl     1  I  615     1      J 

l,*4     1     0  '-     ^  oT6     '2     R 


C.  D  "ns. 

/"f.  ;-/V.  //f.  5'?.  pV.  T'.  ;-'. 

7                      8     1     .;  17    2  5    1 

',     4                      .116  6    '2  6     1 


•'.£. 

.'>'     ...     gt     '.     ,-jf.     flf.  T.  •!.' 

0      I      O  2  >      \ 

.4     *      1      .*>  ]{)     fO     3      I  f      (2     27     0     0 


'.    (ft.  pf.  /?''•'.    «•(;/.   ',».   /''. 

or:    C5    i    n 
75    :>7    i    i  a.-;    s    o 


8.    LO\G 

6.r.  m.  fur.  pn.  fr.  ir.fir.fn. 

11     0  41     6  f(> 

2     2     11      I  ID     G  £4     1     7     SI 


/<•.     m.fur  ]*>.  If.  n.,  fur.  p*.  If.  m.  f 

I     6    »7  l(i     •)'    I     3  9     2  "  0      7 

1Q     2     4      »9  Id     !      3     5  1118 


GO.MPUL'XD    SUBTRACTION.  41 


9-  LAND  OR  SqUARE  MEASURE. 

J).  roods,  rods.  Ji.  r.  po,      5.7.^.  s^.m. 

£3    I    10  29  2  17       S99   131 

£4    1    25  17  I  36        19   15* 


.#.  </?•.  mk.  *#.  yr.  rotls^.  sq.ft.  aq.in. 
540  0  23  130  1  10  860  84 
11!)  1  £7  49  1  H  1-13  125 


10.   SOLID    MEASURE.-* 

••tons.  ft.                »     conls.  ft.  £<>??*.   /?.  •»??. 

llfi   '24                         72     114  45    *18  140 

109     59                        61     i:o  16     14  14* 


11.  TIME.. 

r/rs.  wo.  «•.  r/(7.  yrs.  rfay.c.  ft.  win.  s?c. 

:?-*      II     S     1  '24  S52  20     41     2fl 

43     11     3     5  14  a-5f)  20     4i>     19 


?r.      rf.   A.    min.  SP<V  w.     rf.    j(r.  -ndn.  sec. 

472     2     13     18     44  781     I       8     23     21 

218     4      16     29     54  1 1*7     3      1'2     42     55 


l2.    CIRCULAR   MOTION'. 

9     23     45     54  •  <>     liO     34     54 

S       7     40     56  7     29     40     S6 


59  QUESTION'S,     &G.. 

QUESTIONS, 

Showing  the  use  of  Compuuntl  Addition  and  Subtraction. 

M:\V-YOKK,  MARCH  £L2,  1814. 
1.  Bought  of  George  Grocer, 

19  C.  2  firs,  of  Si!!rar.  at  52s.  per  cvvt.         £52     10     0 

23  lh.«.  of  -i- 1!).  070 

S  lonvos  (if  viiLar.  \vt.  /Uib.  at  Is.  !</.  per  Ib.  1      17  11 

5  C.  2  qrs.  I4lb.  of  liaisins,  at  56*.  per  cwt.    6     10     § 


S.  'What  sum  added  to  17Z.  11s.  SJrf.  will  make  10W.  ? 

.//•..  . 
5.  Borrowed  ~>nl.  IDs.  pakl  again  at tme  time  17'.  11.9. 

9<L  ami  at   aii'jt/'Cr  tin10.   9/.  4s.  '  7J. 

•.d  at  ano/lier  tinir  19.s.  (>i£i.  iio\v  ••  r.ins 

unpaid?  !/?7i.s.  £15  -U.  9jrf. 

4.  Hornnved  !()('/.  ni;d  paid  in  p:1 

oiM-  ri  me  £H.  lls.  6/7.  ;U  anothel*  titne    19/.  171-'.  ^!^.  at 

time 

:>i|    fliil'MS    at  ...  i.'Cils^l't 

.  I'uc'i ;  iii)vv  much  remains  doe,  or 

'//. 

5.  A,  B, and  C,  drew  their  pri'/,c  tni  -:,  \\T~. 

A   had    7.)/.    l.)s.  4.i'.      1  -  A. 

tacking  15s.  6cF.  and  (%  hat)  ji  -thj 
| 

I  !<•  it  I'pfcr  'i  Mils  lent 

bini                                    e.     lit'  lias  paid  me  :u   or.r  tiir.e 

f/  i  d. i'  I.  40  i  .  be- 

s'ulf-  .  ;'.  tor 
]•}.)  dols.  'JO  ct 

' 

7.  r.  .  ios. 

'.W. 
•i  order  <• 

!)«•:• 

to  know  what  stinv^i  uiaxc  ut.  Lui 

"JST:J 


MULTIPLIffiATTOX.  51 

S.  A  merchant  had  •              .  owed  lira. 
S917/.  ICs.  &'.    V. 

9rf.  of  it;  v,      r  .                  \    \(\s.  9tl. 

9.  A  merchant  b<;:<  .'•'  sa^ar,  of 
"which  lie  sells  9C.  3qrs.  2:3lb.  how  much  of  it  remains 
unsold  ?  s-.  i  :-lb. 

10.  From  a  fashionable  piece  of  cloth  which  <•  -'i'aimul 
52yds.  2na.  ataylor  was  o.-dcred  (:<>  take  three  suits,  each 
6yda.  £qrs.  how  uiucii  reiaaius  •.{'  tin-  piece  ? 

»'J    -.      '.yds.  %r«.  2«a.. 

11.  The  \var  between  K<  i  .Vmerica  coalmen-, 
eed  April  19,  17T5,  aur                 J  jjcacc  took  place  Jan- 
uary 20tii,  1785  ;  hw\y  lon^  flitLt  u-  ci/iitiriup  : 

I '/. 


COMPOUND  MULTIPLiCATl     N. 

COMPOUND  Multiplicutlon  is  when  the  Multiplicand 
consists  of  several  denominations,  &i.c. 

1.  To  Multiply  Federal  Muney. 

RULE. 

Multiply  as  in  whole  numbers,  and  place  the  separa- 
trix  a*  many  figures  from  the  li^ht  hand  in  the  product, 
as  it  is  in  the  multiplicand,  or  given  sum. 

EXAMPLES. 

S  cts.  g  d.c.m. 

1.  Multiply  55   09j>y25.    Z   Multiply  49  0  0  5  by  9f. 
25    *  97 


17545  3-T>n35 

7ulB  4-410-45 


Proa'.   g877,  2?  S  5 

S    »/*. 

3.  Multiply  1     cloi.  4  ct5.  by          f^.l  7,  .:<5 

4.  Multiply  41  cts.  5  mills  by 
fi.  Multiply  9     dollars  by 

••tiply  9     Cfifs  by 
7.  Multiply  9    mills  by  50    Jins.      0,  45' 


'T&  COMPOUND  MULTIPLICATION. 

8.  There  were  forty-one  men  concerned  in  the  pay 
ment  of  a  sum  of  money,  and  each  paid  3  dollars  and  9 
wills;  how  much  was  paid  in  ail? 

3ns.  2123  S6cfs.  Smith. 

9.  The  number  of  inhabitants  in  the  United  States  is 
five  million*;  now  suppose  each  should  pay  the  trifling 
sum  of  5  cent*  a  year,  tor  the  term  of  12  years,  towards  a 
to  tinental  tax  j  how  many  dollars   would    be  raised 
fLereby  ? 

three  millions  Dollars. 


2.  To  Multiply  the  Denominations  of  Sterling  Monty, 
Weights,  Measures,  <£'c. 

RULE.* 

"Write  down  the  Mu!ti])licand,  and  place  the  quantity 
4indern<tath  the  'east  denomination,  for  the  Multip'i'-i  , 
and  in  multiplying  by  it,  observe  the  same  rules  for  carry- 
ing from  one  denomination  to  another,  as  in  Compound 
Addition. 

INTHODUCTORIT     EXAMPLES 

£.    s.  d.  q.        ,  5.  </. 

Multiply  1  11  6  2  by  5.         How  much  is  S  times  11  9 

5    '  5 

£7  1J  8  ft  £1    1;»  3 


fc 

«.     r/.                      £.     s.     «. 
10     8                     24     12     6 
*                                     S 

/!  .    .•».    </. 
31     15     S 

4 

^9 

11     10                  10     16    4 

5                                  6 

• 

31      10     9* 

* 

When  accounts  are  kent  in  pounds,  s'.iill 
kind  of  ra.JUplication  is  n  r<mri«o  and  el 
inp  Uie  valur   of  *ooils,  ."i   ^>  much  p«-r  ^ 
ttraJ  i«l«  being  to  multiply  the  ^iren  price 

iiid  ppnc<>. 
inftnod  of 
.11^},  Ib.  tr.  t)i«i 
by  die  quantity 

COMPOUND    MULTIPLICATION.  f>& 

31     16     8  12     17     10  14     10     7$ 

8  9  10 


S2     12     10  6.    19     1  £68     4.} 

11  12  12 


Prr  •','*!  inns. 

"What  cost  nine  yard.,  ofcKitii  at  5s.  6<1.  per  yard  ? 

£0  5  6  price  of  one  yard. 
Multiply  by  9  yards. 

..0ns.  £  2  9  6  price  of  nine  yards. 

QUESTIONS.  ANSWF.ttS. 

£.     s.  rf.  £.     K.  d. 

4  gallons  of  wine,        ai  0     8    7  per  gallon.  1      14  4 

5  u.  Malaga  Raisins,  at  1     2    cj    per  cwt.    5     11  3 

7  reams  of  paper,          at  0  17     9  J- per  ream.  6      4  6$ 

8  yds.  of  broadcloth,    at  1     7     9ipe--yard.il       24 

9  Jb.  of  cinnamon,        at  Oil     4^  per  1ft.        5       2  2J 

11  tons  of  hay,  at  2     1  10    per  ton.    23       0  2 

12  bushels  of  apples,     at  0     1     9   per  bush.    1       10 
12  bushels  of  wheat,      at  0     910    per  bush.    5     180 

2.  When  the  multiplier,  thnt  is,  the  <|ua,itity,  is  a  com- 
posite number,  ami  greater  than  12,  take  any  two  such 
numbers  as  when  multiplier  r,  will  exactly  pro- 

Huce  the  given  quantity,  and  multiply  first  by  one  of  those 
figures,  and  that  product  by  the  other;  and  the  last  pro- 
duct will  be  the  answer. 

EXAMPLES. 

What  cost  28  yards  of  cloth,  at  6s.  lOd.  per  yard  ? 
"  £.  s.    d. 

0    6     10  price  of  one  yard. 
Multiply  by  7 

Produces  2    7     10  price  of  7  yards. 

Muli;;>ly  by  4 

Answer,  £  9     11     4  price  of  28  vjyd?. 

S» 


51  60MPOUV&   iii'LI'IH.ICATiOX. 


4UE*TiOM« 

AXSUTRH 

<e. 

rf. 

.--. 

d. 

24 

£7 

yardi 

at 
at 

/ 
9 

4 

3  per  vard, 
()      —1 

= 

a 

13 

17 
5 

6 

6 

44 

— 

at 

1~ 

_^_ 

=- 

4 

6 

91 

— 

at 

8 

3 

1       — 

= 

14 

H«  i 
-  • 

7'i 

— 

at 

11 

1)       — 

1=3 

71 

14 

0 

SO 

— 

,i 

— 

=i 

3 

10 

10 

84 

__ 

ac 

18 

4 



== 

77 

S 

6 

96 

— 

at 

11 

0       — 

=s 

56 

8 

B 

65 

—  at 

£-1 

17 

6 

0       — 

= 

o 

6 

i-:4 

--    at 

i 

4 

o 

0       — 

=^ 

174 

0 

0 

S.  When  no  two  nuiubers  multiplied  together  will  ex- 
make  die  mu!  two 

C  jinxluct  will  come  tin- 
•line  by  what  remained  ;  \ 
..ict  gives  tlic  utisv. 

LMPZ.ES. 

"VVh.it  will  47  yils.  of"  d«,th  CiJine  tu  at  17s.  9J.  per  vd.  ? 

£.     s.     d. 

0     17    9  price  of  I  v 
Multiply  by  5 

4      8    9  price  of  5  ;  a:  da. 
by  9 


iVuducct   39     18     9  prio 
1     1 

.  £41     11     3  pi-ice  of  47  ya. 

INS.  /vswr 

£.  s\  d.  £.  s.     d.  ' 

"t"  !i:  «iu      at  0     ;,  GJ,  per  t'H.  ^4  1     5J 

17  ells  nl'  ilm-.l.is,  at  0     1  Gj  |.e,-  t-ll.  I  6     2i 

S9  t\vt.  i.                    at  :•>   10  0  prrcwt.  137  9 

j«  vi!*.  of  cUitli,      at  0     5  !*  per  vd.  14  ID     d 

19  Ib*,  of  indigo,    at  0  H  6  per'lb.  M  is     •', 

L  0  13  7  per  yd.  19  ir>  1  : 

lit   vJs-  broadcloth,  at  1     2  6  per  yd.  124  I 

?i  Eteavcr  luta,       at  1     0  4  apiece.  157  IT 


COMPOUND    MUJLTlrLIOATlON.  55 

4.  To  find  the  value  of  a  hundred  weight,  by  having 
the  price  of  one  pound. 


i  f  the  price  be  far!  .icgs,  multiply  2s.  4d.   by  the 
hijrs  iu  the  price  of  '.me  lb.—  Or,  if  tin;  price  be  pe 
7JiuIti;iI\  9s.  -id.  byf  e  j-ence  In  the  price  of  one  lb.  and 


fcXAMPLES. 

What  will  1  cvvt.  of  rice  come  to,  at  2  id.  per  lb.  ? 

s.    J. 
2  farthings«=2    4  price  1  cut.  at  id.  per  Ii>. 

9  fart!.i/^3  in  the  price  of  I  lb. 

»#/<*-.£  1    1     0  price  of  I  cwt.  at  9  -\  per  lb. 
What  will  I  cwt.  oi  lead  come  to  at  ;\>.  j.er  lb.  ? 
s.     d. 
9    4 


J3ns.  £5     5     4 

Questions.  Jnsiyers. 

1  cwt.     at  2i«"  per  lb.  —  £i     3     4 

1  din*,  at  2;-d.  —  =      158 

1  ditto,  at  3d  —  =      180 

1  ditto,  ut  'M  —  =      0  18     8 

1  ditto,  atakl  —  sa      i  IxJ     3 


E.^amplfs  of  Wrights.  Measures,  <J 
1   Howiuucli  ii  j  times  7c\vt.  3qv*.  15  lb.  ? 
CVJ.  iyra-.  /6. 
7      3     15 


JKS.  Cwt.  35       I     19 

//»   ox.  ;m'?.  ^-r.  CM-/,  (/r.  lb.  oz. 

fi.  Multiply  20   2    7      13  by  4.  (3)  2.7     I     13    12 

4    "  G 


IVmluci  !b.  80   9  10       4  lb.  164    0    26    8 


56  COMPOUND    MULTIPLICATION. 

ANSWERS. 

yds.  qr.  nn.  yd*.  <jr.  na. 

4.  Multiply  14     3     2  by  11 

hhd.  g.  qt.  pt.            .  't.  pt. 

5.  Multiply  21   15     2     1  by  12  -.i     0 

I?,  in.  fur.  po.  r.  ;)0. 

fi.  Multiply  81  2  6  21  by  8  CjJ  1  '-t  8 

a.  r.  ;>.  r.  /?• 

r.  Multiply  41  2  11  by  18  748  0  38 

j/r.  7/J.  z.  ;/r.  7».  . 

8.  Multiply  CO    5     3     6  by  14 

^     o      ;     -    '  o      .     „ 

».  Multiply    1      15  48  24  by  5  7     10    2    0 

cds.  ft.  •  ft. 

IP.  Multiply  3    b7  by  8  29    56 


'Practiced  Questions  in 
WEIGHTS  &  MEASIT. 
1.  What  istlic  Moi^it  of  7hlids.  of  sugar,  on 
\Vhat  is  '  •  of  6  chests  of  tea, 

How  much  bra..uy  in  P  ing  41 

-.  1  pt.  ? 
4. 

5.  In  •'  IK!,  and 

. '  u\v  man\ 

f  wood,  i.  and 

96  !  :  .•»  ? 

7. 

. 

21  o^.  15  pv.  *.      1 
v.liole?  4-2«s.  £.'6.  lOo^.  2pu.'t.  * 


COMPOUND    DIVISION.  5f 

COMPOUND   DIVISION, 

1EACHES  to  find  how  often  one  nvunber  is  contained 
ift  another  of  different  denominations. 

DIVISION   OF   FEDERAL    MONET. 

$'3s»Any  sum  iu  Federal  Money  may  be  divided  as  a 
whole  number;  for,  if  dollars  and  cents  be  written  down 
as  a  simple  number,  the  whole  will  be  cents  :  and  if  the 
•urn  consists  of  dollars  only,  annex  two  cyphers  to  the 
dollars,  and  tfee  whole  will  be  cents ;  hence  the  follow- 
ing . 

'  GENERAL  RULE. 

* 

'Writedown  the  given  sum  in  cents,  and  divide  as  im 
whole  numbers ;  the  quotient  will  be  the  answer  in  cents. 

NOTE.  If  the  cents  in  the  given  sum  are  less  than  10, 
you  must  always  place  a  cypher  on  their  left,  or  in  the 
ten's  place  of  the  ceuts,  before  you  write  them  down. 

EXAMPLES. 

1.  Divide  35  dollars  68  cents,  by  41. 
41)3568(87    the  quotient  in  cents  ;  and  when  there 

328  u    any  considerable    remainder,   you 

-•••  -  may  annex  a  cypher  to  it,  if  vo  ;  cii-ase, 

2oS  and  divide  it  again,  and  you  will  have 

287  the  milla,  &c. 

Rem.       1 

2.  Divide  21  dollars,  5  rents,  by  14. 
14)2105(150  cents=i  dol.  Jo  ct«.  bat  to  bring centi 

14  int.- 

fig1';  '.>,  and 

TO  the  rest  will  be  doilais,  &.C, 

rc 

5 

5.  Divide  4  dols.  9  cts.  or  409  cts.  by  6.     tins.  68  c^s.-}- 
4.  Divide  9  dola.  24  eta.  by  12.  *fl;w.  77  cii. 


59  COMPOUND    DIVISION.  ' 

f.  Divide  97  dols.  43  cts.  by  85.      Ms.  gl  l-lctt.  Cm. 

6.  Divide  24b  dols.  54  tts.  b"\ 

JHS.  lOHrfs.  8m.=Sl  OMs.  8m. 
'  T.  Divide  24  dols.  65  cts.  by  '248.  Jns.  9<-fs.  9m. 

8.  Divide  10  dols.  or  1000  cts.  by  25. 
.  9.  Divide  125  dols.  by  500. 
JjD.  Divide  1  dollar  into  33  equal  parts.         J/is.  SoJs.-f- 

PRACTICAL  QUESTIONS. 

1.  Bought  25lb.  of  coffee  for  5  dollars;  \\hatisthat  a 
pound  r 

"2.  If  131  yards  of  Irish  linen  cost  49  dols.  78  cts.  \\hat 
i*  that  per  yard  r  JJii-. 

5.  If  an  c\vt.  of  sugar  cost  8  dols.  96  cts.  what  is  that 
per  pmiml  r  s.  8c/s. 

4.  If  140  reams  of  paper  cost  529  dols.  \vhat  is  ihat 
per  ream  ? 

5.  If  a  reckoning  of  25  dols.  41  cts.  be  paid  equally 
among  14  persons,  \\hat  do  tiiey  pay  a 

!.  gl  81V/S. 

6.  If  a  man's  wajjes  are  235  dols.  8'  .hat 
is  that  a  calendar  month  ? 

7.  T!.o  s.iui-y  of  the  President,  of  the  1'niU'd  :.-r;'.ft's  is 
|»venty-five  tfaovsand  dollars  a  year  ;  v.  hat  is  thai  a  day  ? 

s.  £68 


2.  To  divide  the  (Ipnnminations  n/  Sterling  JIuney, 

' 


RULE. 

vit.h  ^  .,••  cU>nomination  as  ii 

Fthe 
reinainde 

>-:iiation:  llu-n  ihv 


•iidcr,  if  any,  as,  before;  and  bu  on,  till 
PROOF  —  The  same  as  in  Simple  Division. 


(JOMFOUND    DIVISION'.  53 
EXAMPLES. 

.     s.    d.    qr.  £.    s.    d.    . 

te     97     3     12    2  by  5.  8)27     13    6 

Quo't.  £19    892  £3      9    9f 

£.     s.     cL  £.  .v.     <L 

S.  Divide     3!    11     6    by    2  Jlns.  15  15     9 

4.  Divide     22      3     9     by     3  7     7  11 

5.  Divide     70    10     4     bv     4  17   12     7 

6.  Divide     56    11     5J  by     5  116     SJ 

7.  Diude     61    14     8     by     6  10     5     9£ 

8.  Divide     24    15     6^  by     7  3  10     9J 

9.  Divide  185    17     6     bv     8  23     4     &| 

10.  Divide   182    16     8     by     9  £0     6     3.J 

11.  Divide     16      1  11     by  10  1  12    2* 

12.  Divide       1    19     8     by  H  0     3     7i 

13.  Divide      666    by  12  0  10     6£ 

14.  Divide       1      2     6     by    9  0     2     6 

g  15.  Divide  948    11     6    by  12  79     0  11 J 

2.  When  the  divisor  exceeds  1 2,  and  is  the  product  of  tws 
or  more  numbers  in  the  table  multiplied  together. 

RULE. 

Divide  by  one  of  those  numbers  first,  and  the  quotient 
other,  and  the  l;i>t  quotient  will  be  the  answer. 

EXAMPLES. 

£.   s.  (I-  £.  f.tl. 

1.  Divide    29  15  0  by  21  Jlns.  I     8  4 

2.  Divide    27  16  0  by  32  0  17  44 

3.  'Divide    67    9  4  by  44  1    10  8 

4.  Divide    24  16  6  bv  36  0  13  9$ 

5.  Divide  128     9  0  by  42  312 
(5.  Divide  2G9  12  4  by  56  4  16  Si 
7    Divide  248  10  8  by  64  3173 
8.  Divide     65  14  0  by  72  018  3 
*>.  Divide      5  10  3  by  8i  0    1  41 


•0  COMPOUND    DIVISION. 

f.sd  -C.    s     tf. 

10.  Divide  115  10     0  by  90.  1      5      S 

]!.                           16  !>v  KM.  154 

;     G  I»V  Kl.  1    13      6 

13.    .                  4     4     0  by  144.  0      4      S 

r.  When  tli<*  dr. i?,;>r  is  larj:?,  and  not  a  composite 
nu!i::;ei,  you  IH.IV  divide  by  the  whole  divisor  at  once* 
alter  moaner  of  long  division,  as  follows,  vi2. 

EXAMTLFS, 

Divide  V28/.  13.«.  3d  by  47. 
f..    a.     d.  £*     s.     (1. 
47)1*8  13    5(2     14    9    quotient 
94 

54    pounds  rewninmrr. 
Multiply  by    20  and  add  in  the  : 


produces    693  shillings,  which  divided  V 

47  [Ks.  in  the  quotient 

223 
188 

Multiply  by      12  and  add  in  the 

produces    4C.1  pence,  which  divided   as  above,  girts 
[9d.  in  the  quotient. 

£••  *•    rf-                                     £•     '• 

2.  Divide  iffl  15    4    by    51.            A:>. 

3.  Divide     85  6    5    bv    r.^.                     1 

4.  D                ;  5  10 ;.                                    0 

5.  Divide   IW  0     8     liV 

C.  Divi:!«»  740  T 

7.  Divide  88S  13  10    by                            9 


COMPOUND    D1VISIOX. 
EXAMPLES    OF 

WEIGHTS,  ME  AS  I-     ^^^ 
T.  Divide  14  cut.  1  qr.  8  Ib.  »f  si^^^mjr  aroorrg 
6  men. 

C.    0r.  Vo.  ox. 

8)14     1     8     0 

I    S    4    8  Quotient. 
8 


14     1     8     0  Proof. 

«.  Divide  6  T.  11  c\vt.  S  qrs.  191b.  by  -1 
Jns.  IT.  12r;rf.  3r/r 

3.  Divide  14  cwt.   1   qr.  1C  IK  by  5 

Jlns.  Zcvtt.Sfrs.  I3lb.  Dox.  9a7r.~f- 

4.  Divide  16  Ib.  13  ox.  10  dr.  by  6 

Jlns.  -7(7;.  r:Vr.  I5t/r. 

5.  Divide  56  I  b.  Go/,.  lT|nvt.  of  silver  into  9  equal 
parts.  Ans.  6/0.  Soz.  Kpict.  l^grs.-\- 

6.  Divide  26  Ib.  t  07.  5  p\vt.  by  2  I 

J»s.  ilb.  lor.  !/:«•.'.  1, 

7.  Divide  9  hlids.  28  jrals.  2  qts.  by  12 

J«s.  bhhds.  49gols.  2q!s.  1 

8.  Divide  I63bu.  1  pk.  6  qts.  by  S3 

Jlnt.  4bu.  Spkf.  2qts* 

9.  Divide  IT  lea.  1  m.  4  fur.  21  po.  by  21 

.4ns.  2 HI.  4fur.  \po. 

10.  Divide  43.  yds.  1  qr.  1  na.  by  11 

«2».«.  5yds.  Sqrs.  Sna. 

11.  Divide  9rE.E.  4  qrs.  1  na.  by  5 

Jliis.  19yds.  Qqrs.  Sna.+ 

1C.  Divide  4J  gallons  of  brandy  equally  am  on";  144 
solilic.rs.  Jlns.  ]£j-ll  a-pii-sp. 

\\  Bought  a  dozen  of  silver  spoons,  wlrteh  tocetljer 
weired  Sib.  2  07..  15  pwt.  12  grs.  how  much  sihor  did 
•ac!\  spoon  contain  ?  Atis.  Soz.  4pwt.  ll^i*. 

14.  iJn'^ht  17  c'.vt.  5  qrs.  19  Ib.  of  sugar,  and  sold  eut 
eneUiirdui  itj  how  much  remains  unsold  ? 

.  llcwt.  3m, 


§S  COMPOUXD    DIVISION. 

15.  From  ajArce  of  cl^th  r  v.; \Vs   2  na. 

a  r;  jfllfc8^  *:|  '' 

tool.  A      Bhe  *vhoL  ^i\v  many 

C;Ull  Cfll  .    %!*S.    i 

PR  110NS. 

1.  If  9  vrrds  of  ck>th  cost  4Z.  5s.  7i^.  'at  per 

f..  s.   rf.    <7r. 
9)4    3     7    2 


2.  If  11  tons  of  'nay  cost£3/.  ' 
ton  ? 

3.  If  I12  gallons  of  brandy  cost  4/.  15s.  (jr.  -Jiat 

7s.  1 1 .7. 

4.  If  84  Ibs.  ofchcese  cost  ll.  16s.  (•  per 
•',  ? 

5.  Jioiv  s  of  stockings  for  11^. 
a  pair  dn  I                '1  me  in  ? 

6.  I!  ;  51.  8s.  lOi-J.  b?  rai 

v  a-'t)iere  r 
.  contuiai: 
;•  yard  ? 
.;.l   of  vine  ci 
gallon  ? 

0.  If  1  c\\  t.  of  sugar  cost  S/.  IDs.  u hat "«  U 

^       *' 

10.  If  a  man  sjionds  7ll.  14s.  CJ.  a  year, 

ptrcaliMit!;: 

11.  't  Wales'  s 
ifi  Liuf  a    (1 

' 

ii  the  ow 

am1  the  rmiaindcr  is  equally  oi\ : 
ar«  liJ  ia  number  j  lio\v  m\«t 


REDUCTION.  »     G3 

^.«j. 

13.  Three  merchants.  A,  .15,  .and  t^»ve  a  ship  in 
tompany.     A  hath  -*,  B  £,  arid  C   •.  aruM*'  for 

freiht  2281.  ILS.  8d.     It  is  reu:  ide  it  amon 


the  owners  according  to  their  respecthe  s 

dns.  J's  share  £143  Os.  5d.  11'*  share  £  57  4s.  &f. 
C's  s/Mzre  £28  12s.  id. 

}•'.  \  [Jrivateer  ha\ing  taken  a  prixc  wnrtli  £GS50,  it 
is  divideil  into  one  hundred  smarts:  of  which  the  cap- 
tain is  to  have  11;  2  lieutenants,  each  5;  lli  tnidship- 
men,  each  2  ^  and  the  remainder  isttibe  divided  equally 
amon^  the'  .sail  •;«  lijj  in 

J.'2i.  Capt-iin's  share  §753  5!)cf>'.  ttezrf's,  ^3-4  i!  50cf*. 
a  midshipman.'  a  5S137,  aji.'t  a  sa/'f. 


REDUCTION, 

TEACHES  to  bring  or  chains  HUT..!   rs  frori  one  name 

. 

to  another,  without  altei ; 
Reduction  i,  either  LK- 
Descend; 

small,  as  pounds  into  '— 

,,,,  .    .  .      .  -   ,  >  -uion. 

Inis  is  done  by  .Vliiltn  «    .   , 

•    Ascendin<ris  whei    "'      U:l:l'csarc ' 

as  shilling  Into  p<io^  hours  into  «•>•>««- 

performed  by  Uiv'  ' 

RE  UCTKiX  DESCENDING. 
RULE. 

Mu!t:nlv  i'6  '"^iest  ^""'ojnaticn  c-vpn,  bv  so 

of  1 

M.J   !i;»ve  Ui-i.:.;^ht  it  do.- 


' 


-  erclcroftheo 

1 

KXY.MVLKS. 

^  r,,  ,  -•  15-s-  2"-  2qrs.  how  many  f 


rf.  qrs. 

Proof. 
4)24758 

515  shillings.  12)6189 

12  ^_ 

£{0)5 115  9d. 

61 S9  penca.  

.4  £25  15  9| 

-4758  farthings. 

NOTE.     In  multiplying  by  20, 1  added  in  the  15s.-4>f 
the  Od.-uud  by  4  the  Sqrs.  which  must  always  be 
done  in  like  cases. 

'Ms.  lOd.  1  jr.  how  many  farthings  ? 

«    Tn  4«'    T*   \  •  50329 

t:<L  Syrs.  how  many  far  things  = 

4.  In   OIL  12s.  how  ^  ;47. 

thinr-s  -  :;rs'  !HM1(-1-  ami  fu. 

o.°Isi  84/.  how  many  sli  "'Jl36,;r>-. 

J  -i  pence  ? 

6.  In  lSs.9</.  how  many  piv.  'Ji''- 

7.  la  S12/.  8s.  5tZ.  how  many  lialf-p,  :<ITS- 

8.  In  846  Jv»!!:u-s  at  6i.  each,  how  m:-. 

farthings? 

9.  In  41  guineas  at  28s.  each,  how  iv 

•^vnce  r 

10.  Tn  59  pistoles,  at  &2s.  how  inanvshi. 
aud  fa 

Jns.  1 2  'oU 

11.  Iti  37  half-johannts,  at  -18s.  how  P.U 
six-pi 

i.  7104  //<rfc- 
I ..  .it  6s.  8rf.  car1,, 
.dliirthin/.-  ^«».  9ty  iiany 


REDUCTION.  65 

REDUCTION  ASCENDING. 
RULE. 

Divide  the  lowest  denomination  given,  by  so  many  of 
that  name  as  make  one  uf  fee  next  i-iigi'-e;-,  and  so  on 
through  all  the  denominations,  .is  fur  as  jour  question 
es. 

FKQOF.     Multiply  by  the  several  divisors.  ! 

lAMI'LES. 

1.  In  £24765  farthings,  how  many  pence,  shillings  and 

pou  i 

Farthings  in  a  {jenny  =  4)324765 


Fence  in  a  shilling        =  12)56191   1 

Shillings  in  a  pound     =  2jO)468|2  7d. 

'£234    2s.  7d.  Iqr. 

..  5619 Irf.  <i6S2s.  2S4J. 

NOTE.  The  remainder  is  always  of  the  same  name  as 
the  div'nii 

2.  Bring  30329  farthings  into  pounds  ? 

Jus.  £31  11s.  I0d.  Iqr. 

3.  In  44447  farthings,  how  many  pounds  ? 

Jlns.  £46  5s.  lie/.  Sqrs 

4.  In  59136fartlu!i<;.s,  how  many  pence,  shillings,  and 
pounds  ?  JHS.  t4784rf.  1252s.  £61  . 

5.  In  201CO  pence,  how  maay  shillings  and  pou; 

s.  iGBOs.  or  £c4. 

6.  In  900  farthings,  how  many  pounds  f 

.  £0  18s.  Drf. 

7.  Bring  749S1  half-pence  into  pound*  ? 

Jna  £156  4s.  £^7. 
•    8.  In  243648  farthings,  how  many  Collars  at  &».  each  ? 

.<?ws.  g: 
0.  Reduce  15776  pence  to  gtfi^eas,  at  28s.  per  guinea. 

5.    41 

10.  In  C23C4  farthings,  ^ow  many  pistoles,   at  22a. 
sach  ?  * 


II.  In  7104  thrce-pences,  how  roanr  hakf-johannes,  at 

Jus.  S7. 

.•2.  In  SS720  fanhLi  >,  hoy/ mauj  French  crowns,  at 
6s.  £d.  ?  .ftu.  181. 


Reduction  Ascending  and  Descending. 
1.    MONEY. 

I.  In   1C  I/.  Os.  9£d.  how  many  half-rvrro  P 

B09f 
£.  In  58090  half-pence,  how  many  pouri 

»V/;.s-.   l-l/.   Oi 
S.  Bring  23760 half-pence  into  pounds.  viVs.  £-J9  10s. 

4.  In    21-I/.  Is.   S(!.  how  many  sliillin..  '.ccs, 

sia-- 
;;c'?s,  «;ui  2ft550Q  farthings, 

5.  In   t  ;  e,  and  Kn»1is!i  or  French 

••h?  .  .  -MS. 

6.  :  .  ho\v  many  pence  anif 
tig?  .«)«/.  «,:j? -_ 

7.  -v;  laiinv 
gnw 

b.  [n  48  guineas,  <  -4  :d. 

9.  In  81  gv.iucaSj  at  xirs.  4d.  each  how  ir.  .Is  ? 

ii)   145. 

10.  In    C450G  ponce,  }u»\v  .  -;nd 

.  .:S  ? 

0  '. 

II.  T.i  ^i-2  nioitloriv*,  at  ;">t*».  Ciicli.  i 
at  2! 

1  \  Li  !••./) Dutch  guilders, at  -;.nj 

1;'j.  i' 
how  many  ri- 

14.  In  50J.  how  many  \iul!in;.>.  nin*' -:• 
•CS,  tbar-penct's.  ;MU<  ;;••  i»er  ? 

-i./.  ^  v.  -.'.'  ,<,,,/  y;.-;o  =, 


EXAMPLES     IN 

REDUCTION  OF  FEDERAL  MONEY. 

T.  Reduce  £745  dollars  into  cents. 

£745  dollars  1      Here   1  multiply  by  100,  the 
100  1  cent*  in  a  dollar:  but  dollars  are 

>readi!y  brought  into  ceutn 

JUns.  274500  I  noting  twj    ciphers,    and"  into 

-• •  J  mills  uy  annexing  three  cyphers. 

Also,  any  sum  in  Federal  money  may  be  written  dov. :-. 
a-»  a  whole  [lumber  and  expressed  ill  it<  !;>•>•.  • 
tion :  f>r,  wlieti  dollars  and  cents  are  joined  t> 
a  whole  number,  without  a  sepnratrix,  they  viil  ? 
how  many  cents  the  given  sum  contalna:  and  vlien  dol- 
lars, cents,  and  mills  are  so  joined   £ogetoer,  tiiey  \vi'l 
shew  the  whole  number  of  mills  iii  ii.e  given  sum — 
Hence,  properly  Bpeakin^  there  is  no  reduction  ^ 
money  :  tor  cents  are  readily  turned  into  dollars  by  cut- 
ting oii'  the  two  ri^lit  hand  liinirt.-,  and  nulls  by  poiii'ir..- 
off  three  figures  with  a  dot :  leit'nand 

of  the  dot,  are  dollars  j  uud  the  figures  cut  oft'  are  CCJHS, 
vr  cents  and  mil  Is. 

~.  In  34J  dollars,  how  man.  '  ;1  mills? 

•  :tft. 

3.  Reduce  48  do1.-;.  78  cts.  i:::,, -ent.s.         *i».s. 

4.  Reduce  £5  dols.    8  cts.  ii;to  c  .>.  xlo(>S 

5.  Reduce  54  dols.  S6ct^.  5m.  iu  to  mills.  »J/-s.j 

6.  Reduce    9  dols.'9  cts.'9m.  into  mills.    J/is. '. 

g    cts. 

7.  Reduce  41925  cents  into  do!!a.~.      A    . 

8.  Change    4896  cents  into  doll.irs. 

9.  Change  45009  cents  into  doiiavs. 

10.  Bring       4625  mills  into  dollars.  •;  02  5 


2.    THOY    AVKIOKT. 

1.  How  many  grains  in  a  silver  tankard, 
lib.  11  Oi.  15  pwt.? 


63  REDUCTION. 

if),     or.  ; 

I      It      15 
12  ounces  in  a  pound. 

ounces. 
ennwelV-is  in  one  ounce. 


4r.)  pern: 

ins  iu  one  penny  v/c" 


950 


J.    8  rains.     *3ns. 

15  pu-t. 

:  '  .-.  11  oz.  15  p\\t. 


2.  In  l--!o  oz.  how  many  p- 

•i.  U808P. 

5.  V  -.8 

4    i;.  :uany  pounds? 

.     .;•?'.•?. 
.-.-f  gold,   i  -;v.  t. 

1  |)wt.  of  silver,  li'jw  inair^  table 

pvv't. 

.al  luu-bei  of  each 

.    . 

.  ju/.     Tlie;efi>re  23544 


3.    AYOIIIDUI'OIS   AVKIGHT. 

!b.  12  oz.  liow  many  ounces  ? 
[Canie 


REDUCTION. 

569  quarters.  Proof. 

fi8  16)101068 

2bTG  28)1006C 

719 
_  ,  4)359  14ft). 

10066  pounds. 

16  89c«rt.  S^rs.  Ulb. 

60598 
10067 

1G1068  ounces.    Answer. 
SI  In  19  Ib.  14  oz.  11  dr.  how  many  drams  ? 


s.  5099. 
•S.  In  1  ton  how  many  drams  ?  *5ns.  57S449. 

4.  In  24  tons,  17cwt.  Sqrs.   17lb.  5  oz.  how  mir.j 
ounces?  Jins.  892245. 

5.  Bring  5099  drams  into  pounds. 

.  19M.  14or.  11  dr. 

6.  Bring  573440  drams  into  tons.  tins.  1. 

4 

7.  Bring  892245  ounces  into  tons. 

»S.'is.  24  /on*,  17e?rf.  Syrs.  17Zft.  . 

8.  In  12  hhds.  of  sugar,  each  11  cwt.  2.»lb.  how  many 
pounds  r 

9.  In  42  pigs  of  lead,  each  weighing  4cv.  t.  r 
many  father,  at  19c\vt.  2nrs,  ?     .^j»s.  IQfotki 

10.  A  gentfeinan  has  20  hhds.  of  tobacco,  e 
Sqrs.  14lb.  and  wishes*  tw  put    it  info  boxes  c- 

.  «ach,  I  demand  the  number  of  boxes  he  must 

A*s. 

4.    APOTHECARIES'  WKTGIIT. 
1.  In  9!f^  8^  13  2^  I9grs.  how  many  grains. 

.071*.  55799. 
&  'I*  55T99  grains,  how  many  pounds  f 

,«n».  9fe  83  13  29  I9»r. 


i 

TO  REDUGTIOJT. 

5.   CLOTH*  MKASUKE. 

1.  In  95  yards,  h<>\\  many  quai  tors  and  nails  ? 

•fir.  •  Ml. 

2.  In  541  vardi,  oqrs.  Imi.  how  many  n;; 

<.  5469.  . 

3.  In  5783  nails,  how  many  vakils  ? 

Is.    Iqr.  Sna. 

4.  In  61  Ells  English,  how  many  quarters  and  nails  ? 

>na. 

5.  In  56  Ells  Flemish,  how  many  (|i:artci>  and  jiuils? 

LML 

6.  In  148  Ells  English,  how  ma;iv  Ells  rlemi 

rs. 

7.  In  1920  nails,  how  many  yards,  Ells  Flemish,  and 
Ells  English  ? 

.  IfipJB.  F.  find96£.E. 

8.  Flow  many  coats  can  Li  -.  of 
broadcloth,  aliowiti-  l^  yards  to  a  cuut  ?                 .  xil. 

C.    DRY    MF.ASVHK. 

1.  In  136bushel."  n'i arts  and  pints  ? 

2.  In49  bush.  SpkSr  5qts.  hu\v    ma:ivt;ii:. 
3..  Iii  -;:.r04  }.inK  how  many  bu>V.d-,  ? 

9 

5.  '  i%!«r, 

". 
1.  .  gallon! 


i/iilt»  OJ'    > 


KEDt/CTIOS.  71 

5.  In  1T89  quarts  of  cider,  how  many  barrels  ? 

.  146/s.  25qts. 

6.  What  number  of  bottles,  containing  a  pint 

half  each,  can  be  filled  with  a  barrel  of  cider?  «!/ns.  168. 

7.  How  many  pints,  quarts,  and   two  quarts,  each  an 
equal  number,  may  be  filled  from  a  pipe  of  wine  ? 

Jins.  144. 

8.  LONG  MKASUHK. 

1.  In  51  miles,  how  many  furlongs  and  po' 

*.(>  poles. 

2.  In  49  yards,  how  many  feet,  incites,  and  barley- 
corns r  .  14rjV.    . .  rr292*.c. 

3.  How  many  inches  from  boston  to  New-York,  it 
being  248  miles  ?  Jn.-;.  U7l3280i»ic/t. 

4.  In  43 5-2  inches,  how  many  yards  ? 

.  120?/rfs.  2/6.  Sin. 

5.  In  68.2  yards,  how  many  r 

x2-r-ll  =  124?-oft's. 

6.  In  15840  yards,  how  many  miles  and  leagues? 

3lea. 

7.  How  many  times  will  a  oarrif^  "'hoci,  16  feet  and 
9   inches  in    oircu  i-.l   in  ^oi.'itr  fi-jin 
New-York  to  Phila 

.^;.; 

8.  ITov/  many  bar'. 

it  being  SCO  degrees  ?  Jins.  47o 

9.    LAND 

1.  In  £-U  a 

rods  or  perches  r 

.2.  In  C2JG9~  scjiiare  poles,  how  iria;v. 
j 

S.  li  apiece  oT  /.-M!  ;in  -in 

of  17  ;ic:-cs,  3  roods,  an.  :;.utoi'  it, 

many  perciie^are  therein  the  re  •. 

4.  T  ;  contain,  the  '' 

aero  acr.-.s,  I  rood  :  in>.  -  c.aa 

Iliey  bo  duided  irito,  each  s':  >>  r.-ds  ? 

s.  61  shares,  and  44  rods 


ftEDUCTIOW, 
10.    SOLID     MKASl'RE. 

1.  In  14  tons  of  hewn  timber,  how  many  solid  inches*  f 

.    l4xo(>x!r'>8=il'20%(W>. 

2.  lu  19  tons  of  round  timber,  how  nianv  inc' 

13I55SO. 
5.  In  21  cords  of  wood,  ho\v  many  solid  feet  ? 

4.  In  12  cords  of  wood,  how  man  v  .-olid  terta;id  inc' 

JH*.  163-  '.;•>.  and 

5.  In  4603  solid  feet  oi'  wood,  Iv.nv  nuuivcoiV 

* 


I  1.      ! 

1.  In  '!  .  hours,  minutes,  and 

seco;; 

B888A.  41  f 

2.  .' 

4.  I.  ,    how  ma: 

.  nfi.anvda- 

.       .  -.  6  ho'U'H. 

JNS. 

bow 

•^ur* 
• 

7.  .  to  Nover  .  ia- 

-,  huw  in  Jj. 

'•  I.  ' 

» 
. 
2.  Brin:, 

.  13°  25r 

qr  = 
1.  TP  t.-.liof  I'.; 

.-..MCJls   Ut 


REDUCTION;  73 

2.  Borrowed  10  English  guineas  at  28s.  each,  and  24 
English  crowns  at  Gs.  and  8d.  each;  how  many  pistoles 

each  will  pay  the  debt  ?  wins.  £0. 

3.  Four  men  brought  each  17/.  10s.  sterling  value  in 
gold  into  the  mint,  how  many  guineas  at  21s.  each  must 
they  receive  in  return  ?  ctfns.  66  %nin.  14s. 

4.  A.  silversmith  received  three  ingots  of  silver,  each 
weighing  23"  ounces,  with  directions  to  make  them  into 
spoons  of  2  oz.  cups  of  5  <>•/,.  salts  t/f  1  07,.  and  snuffboxes 
of  2  <;z.  and  deliver  an  equal  number  of  each  ;  what 
the  number,?  Jus.  &  of  cacti,  and  I  oz.  >  . 

5.  Admit   a   ship's  cargo   front  Bordeaux  to  i 
pipes,  ISO   hhds.  and  150  quarter  casks  [i  hhds.~| 
many  gallons  in  all ;  allowing  every  jnnt  to  be  a  p;> 
what  burden  was  the  ship  of.''  »flns.  444  i 

the.  ship's  bunlen  was  153  tons.  12e.; 

6.  In  15  pieces  of  clot!!,  each  piece  20yds.  how 
French  Ells  ? 

7.  In  1C  bales  of  cloth,  each  bale  12  pieces,  and  eactt 
piece  25  Flemish  Ells,  how  many  yards  ?     .Int.  2250. 

8.  The  forward  wheels  of  a  waggon  are  14^  feet  in 
circumference,  and  the  hind  • 

many  more  times  will  the  forward  wheels  turn  r 
the  hind  wheels,  in  running  from.  Boston  to  lNe\v-V 
it  being  248  miles  ?  shi?,.  7167. 

9.  How  many  tiir.o^  will  a  .ship  97  feet  6  inches  long, 
sail  her  length  m  the  distance  ot  12800  leagues 

yan'  Ans.      . 

10.  The  sun  is  95,000,000  • 
and  a  cannon  ball  at  ib 

::ou  hall  be,  -"t  tiiAt 
rate  in,  flying  ft  •  tin  r 

Jns. 

11.  The  Sun  travels  tin  •  us  of  the 

half  a  year;!  luteg  and  secondsl 

.3«5."l863^      !08.. 

•kes  does  a 

J*w 

13.  How  lon^  will  it  take  to  couir  am:  ; 

.V?:s.S03A.  20rr"  or  larf  "Z\i:.  -'•)  m. 


FRACTIONS 


14.  The  national  debt  of  England  amounts  to  about  £"9 
millions  of  pounds  sterling  ;  how  long  would  it  take  to 
count  this  debt  in  doHars  (4d.  6d.  sterling)  reckoning 
\vithout  intermission  twelve  hours  a  day  at  the  rate  of  50 
dollars  a  minute,  and  365  days  to  the  year  r 

dns.  94  years,  1*34  days,  5  hours,  £0  min. 


FRACTIONS. 

JT  RACTIONS,  or  broken  numbers,  are  expressions  for 
ajiv  assignable  part  or  an  unit  or  whole  number,  and  (in 
general)  are  ol  two  kinds,  viz. 

VULGAR  AND  DECIMAL. 

A  Vulgar  Fraction,  is  represented  by  two  numbers  pla- 
ced one  above  another,  with  a  line  drawn  between  them, 
thus,  3,  f,&c.  signifies  three-fourths,  live-eights,  &c. 

The  figure  above  the  line,  is  called  the  numerator,  and 
that  below  it,  the  denominator, 
Tl  5  ^  Numerator. 

1  IUS'      J  {^Denominator. 

The  denominator  (which  is  the  divisor   in  division) 
•hows  iiow  many  parts  the  integer  is  divided  into  :  and  the 
numerator  (which  is  the  remainder  after  divi*iou)  t' 
how  many  of  those  parts  are  meant  by  the«iractioii. 

A  fraction  is  said  to  be  in  its  k-^st  or  lowest  terms, 
when  it  is  expressed  by  the  lea>t  numbers  po.oibic. 


when  reduced  to  its  lowest  terms  will  be  i,  and 
equal  to  J,  &c. 

PROBLEM  1. 

To  abbreviate  or  reduce  fractions  to  their  lowest  k 

LE. 

Divide  the  terms  of  the  given  fraction  by  any  number 

rhich  will  divide  them  without  a  romaind 

icnts  again  in  the  same  . 
that  there  is  no  number  greater  than  i  . 
them,  and  the  fraction  will  be  in  if 


FRACTION'S.  75 

EXAMPLES. 

1.  Reduce  ££$  to  its  lowest  terfns. 

8  144  J9J?-  »  tl>e  Answer 

2.  Reduce  ^ff  to  its  lowest  terms,        rfnsuvrs  $ 
S.  Reduce  f^f  to  its  lowest  terms.  $ 

4.  Reduce  7\ST  to  its  lowest  terms. 

5.  Abbreviate  $%  as  much  as  possible.  4i 
(\  Reduce  ||-|  to  its  lowest  terms.  14 

7.  Reduce  -i-f*-  to  its  lowest  terms. 

8.  Reduce  •£-£#  to  its  lowest  terms. 

9.  Reduce  -]-|J-  to  its  lowest  terms.  ^' 
10.  Reduce  |4-r|  to  ^s  lowest  terms. 

PROBLEM  II. 

To  find  tlie  value  of  a  fraction  in  the  known  parts  of 
tlie  integer,  as  to  coin,  weight,  measure,  &c. 

RULE. 

Multiply  the  numerator  by  the  common  parts  of  the 
integer,  and  divide  by  the  denominator,  &c. 

EXAMPLES. 

What  is  the  value  of  f  of  a  pound  sterling  ? 
Numer.    2 

20  shillings  in  a  pound. 

JQenom.  S)40(13s.  4rf.  Ans. 
3 

10 
9 


3)12(4 

12 

t.  What  is  the  value  of  •£§  of  a  pound  sterling  ? 

Jns.  18s.  5d.  Zfy 

8.  Reduce  $  of  a  shilling  to  its  proper  quantity. 

Ans.' 

4.  What  is  the  value  of  |  of  a  shilling  ?      Jlr 

5.  What  is  the  value  <;f  if  of  a  pound  troy  ? 


76  TKACTIOXS. 

6    Ho\v  rri'ict!  is  70r  of  an  hundred  \veight  ? 

Sqrs.Tlb.  W-^oz. 

7.  What  is  the  Value  of  ;  of  a  mile  r 

.  (fur.  £6^0.  lift. 

8.  How  much 

.  3y»v.  S/&.  lor.   Il2*dr. 

9.  Red'-  .  fcs  proper  quantity. 

iln*-  ,na. 

of  ahlul.  of  v  ^nJ. 

5  5  A-  sec. 
PROBL 

To  red  e  fraction  of  any 

•>f  the  same  kind. 
'.E. 

Tied'.  -t  term  mention* 

el  i-  •> the 

ror:  ^.vhich  uill  be  the  frac- 
tion required. 

,Bg, 

1.  Reduce  15  •  und. 

£0  Iiiiogral  {/art  ven  sum. 

|0 

4  4  ' 

N'una.     .7, 

Si.   v>  !!>.  ? 

rcm| 

.     iia.  ?  .4j'- 

4.  ^ 

5.  ^s 

•  •--•& 
ft.  ? 

7.  Ri'ilu,  • 

8.  What  jwt  of  au  acre  u  2  roods,  20  pu! 


9.  Reduce  54  gallons  to  the  fraction  of  a  hogshead  of 
wine.  Jlns.  f. 

JO.  What  part  ef  a  hogshead  is  9  gallons  ?     Jtos.  | 
1  1.  What  part  of  a  pound  troy  is  lOoz.  lOpwt.  IQgrs.  ? 

Jiu.  ft* 

BECIMAL  FRACTIONS. 

A  Decimal  Fraction  is  that  whose  denominator  is  an 
unit,  with  a  cypher,  or  cyphers  annexed  to  it,  Thus,  -/y, 


The  integer  is  always  divided  either  into  10,  100,  1000, 
&c.  equal  parts  ;  consequently  the  denominator  of  the 
fraction  will  always,  be  either  10,  100,  1000,  or  10009,  &c. 
which  being  understood,  need  not  be  expressed  ;  for  the 
true  value  of  the  fraction  may  be  expressed  by  writing 
the  numerator  only  with  a  point  before  it  on  the  left  hand 
thus,  T«j,  is  written  ,5  ;  TyT  ,45  ;  j-ffo  ,725,  &c. 

But  if  the  numerator  has  not  so  many  places  as  the 
denominator  has  cyphers,  put  so  many  cyphers  before  it, 
viz.  at  the  left  hand,  as  will  make  up  the  defect  ;  so  write 
yfo  thus,  ,05  ;  and  rf^  thus,  ,006,  &c. 

NOTE.    The  point  prefixed  is  called  the  separatrix. 

Decimals  are  counted  from  the  left  towards  the  right 
hand,  and  each  figure  takes  its  value  by  its  distance  from 
the  unit's  place  :  if  it  be  in  the  first  place  after  units,,  (or 
separating  point)  it  signifies  tenths  ;  if  in  the  seeond, 
hundredths,  &c.  decreasing  in  each  place  in  a  tenfold  pro- 
portion, as  in  the  following 

NUMERATION   TABLE. 


0  C     • 

as   rj    f 
a:   ai  -n   i* 


7654321  254567 

Numbers. 


78  DliCHuAL   TRACTIONS. 

Cyphers  placed  at  the  right  hand  of  a  decimal  fraction 
do  :f.)t  alter  its  \o.lue,  sitoce  every  significant  figure  con- 
tinui  'is  the  sa:np  place:  so  ,5  ,50  and  ,500 are 

all  I'  fa  or  i- 

But  cyphers  placed  at  the  U'.l't  hand  of  decimals,  de: 
cvca  :!ue  in  a  tenfold  pi.^ortioii,  by  minting 

them  furti'.;1!-  i'ro.n  the  decimal  pi-Hit.     'I'hus,  ,5  ,05  ,GU5, 
&c.  .:  hundredth  p;i.  thou- 

saiM  i'espoctively.     It  is  therefore  evident 

that  .Hide  of  a  decimal  fraction,  compared  with 

id  upon  the  number  of  its  figures, 
alue  of  its  first  left  hand  figure:  tor  in - 
atar-  .  any  figure  less  than  .9 

sue! i  c.  if  extended  to  an  infinite  number 

of  figures,  will  not  equal  ,9. 


ADDITION  OF  DECIMALS. 
RULE. 


1  .  Plnrc  the  nn-i'bcr^,  \\hether  mixed  or  pure  decimals, 
under  each  other,  recording  to  the  value  of  their  places. 

2.  Fi;ifl  tlieir  r.ii-.i  ns  ia  v.hole  numbers,  and  poii. 
BO  many  place-  'ccimals.  as  are  equal  to  the  great- 

est number  of  decimal  parts  in  any  of  the  given  nuiv.'- 


EXAMPLES. 


1.  Find  the  sum  of  41,G53-f-S6,05-f  24,009-f  1,6. 

.  '53 


&'>'  \  parts  of 

.un- 

nr  1. 

RAL 

;ie  law  of  notation, 


DECIMAL    TRACTIONS.  .  J 

For  since  dollar  is  the  money  unit;  and  a  dime  being 
the  tenth,  a  cent  the  hundredth,  and  a  mill  the  thousandth 
part  of  a  dollar,  or  u:uf.,  it  is  evident  that  any  number  of 
dollars,  dimes.  d  mills,  is  simply  the  expression 

of  dollars,  and  ;  \irts  of  a  dollar:   Thus,  11  dollar*, 

0  dimes,  5  cents, =11, 65  or  HT6/o  dol.  &c. 

2.'  Add  the  following  mixed  numbers  together. 

(2)                           (3)  (4) 

lards.  Out.  Dollars. 

46,23456                       4o6  48,9108 

24,90-400  7,891  1,8191 

17,004 11  2,54  3,1030 

3,01111  5,6  ,7012 


5.  Add  the  following;  sums  of  Dollars  together,  viz. 
SI  2,345  65  +  7,891-1-2,34 +14,-f  ,0011 

-  An*.  S36.57775,  or  gS6,  5dL  Tcts.  T^mills, 

6.  Add  the  following  parts  of  an  acre  together,  viz. 

,7569 +,23 +,654 +,199 

Jlng.  1,8599  acres. 

7.  Add  72,5-f  32,071 4-2,1574+371,4+2,75 

Ans.  480,8784 

8.  Add  30,07+200,71+59,4+3207,1 

Ans.  3497.28 

9.  Add  71,467+27,94  +  16,084+98,009+86,5 

Ans.  300 

10.  Add  .75C9+,C074+,C9+,8408+,6109 

Ans.  2,9 

11.  Add  ,6+,099+,37+,905+.  Ans.  2 

12.  To  9,999990  add  one  millionth  part  of  an  unit, 
and  the  sum  will  be  10. 

1.5.  Find  the  sum  of 

Two::'  ._.... 

Three  hundred  and  Mxty-five  thoysandths, 
Six  teiitlis,  and  nine  Hitllionths,      -    -     -     - 


Answer,  1,215009 


CO  DECIMAL   TRACTIONS. 

SUBTRACTION  OF  DECIMALS, 
RULE. 

"Place  the  numbers  according  to  their  value ;  then  sub- 
tract .is  in  whole  numbers,  and  point  off  the  decimals  as 
ix  Addition. 

EXAMPLES. 

Dollars.  Inches. 

I.  From  125,64  2.  From  14. 

Take    95,58750  TaJce    5,91 


S.  From  761,8109          719,10009          27,15 
Takte     18.9113  7,121  1,51679 


6.  From  430  take  245,0075  ,?H.V.  £54,9925 

7.  From  23G  dols.  take  ,549  dols.     Jns.  &235,451 

8.  From  ,145  take  ,09634  Jrw.  ,04816 

9.  From  ,2754  take  ,2371  .7ns.  ,0383 

10.  From  271  take  215,7  Ms.  55,3 

11.  From  270,2  take  75,4075  J*s.   194.7925 

12.  From  107  take  ,0007  -.  106,9993 
15.  From  an  unit,  or  1,  subtract  the  millionth  part  of 

itself.  .  ,9995)99 


MULTIPLICATION  OF  DECIMALS. 
RULE. 

1.  Whether  they  be  mixed  numbers,  or  purr  <!<v 
place  the  farfors  and  multiply  them  as  in  whole  nun  ••'• 

2.  Point  off  tvo  many  livv.n  >  from  the  pn><; 
are  decimal  places  in  botii  the  I  actors  :  and  it"  * 

not  so  many  places  in  tlie  p'-odurtt  supply  the  defect  K» 
Vefixidg  cjpucrs  to  tJ»e  left  hand. 


DFCIMAL   FRACTIONS,  81 

EXAMPLES. 

I.  Multiplv  5.  2.  Multiply  S,0fi4 

by    ,008  by 

Product     ,041888  6,74552 

3.  Multiply    25,233  by  12,17  Answers.    307,14646 

4.  Multiply        .2461  bv  ,:529  130,1869 

5.  Multiply        785S  by  3,5  27485,5 
B.  Multiply  ,007853  by  ,035  ,000274855 

,004  by  .004  ,000016 

8.  What  cost  6,21  yards  of  cloth,  at  2  dols.  32  c- 
mills,  per  yard?  ,1ns.  $14,  4d.  5c.  B-^.m. . 

9.  Multiply  7,02  dollars,  by  5,27  dollars. 

.-..  36,9954rfote.  or  g36  99cfs.  5-fom. 

10.  Multiply  41  dols.  25  cts.  by  120  dollars. 

Ans.  g4950 

II.  Multiply  3  dols.  45  cts.  by  16  cts. 

Ans.  S0.5520==55cfs.  Zmills. 

12.  Multiply  65  cents,  by  ,09  or  9  cents. 

x.  gO,0585=5cte.  Squills. 

13.  Multiply  10  dols.  by  10  cts.  Jins.  gl 

11.  Multiply  341,45  dols.  by  ,007  or  7  mills. 

Ans.  g2,39-f 

To  multiply  by  10,  100,  1000,  &c.  remove  the  separa- 
ting point  so  many  places  to  the  right  hand,  as  the  mul- 
tiplier has  cyphers. 

f  Multiplied  by  10,  makes  4,25 

vSo  ,425 -j by  100,  makes  42,5 

(^ by  1000,        is  425, 

For  ,425x10  is  4,250,"  &c. 


DIVISION  OF  DECIMALS. 
RULE.  . 

1.  The  places  of  the  decimal  parts  of  the  divisor  and 
Quotient  counted  together,  must  always  be  equal  to  those 


82  DECIMAL   FRACTIONS. 

in  the  dividend,  therefore  divide  as  ift  whole  numbers, 
and  from  the  right  hand  of  the  quotient,  point  off  so  ma- 
ny places  for  decimals,  as  the  decimal  places  in  the  divi- 
dend exceed  those  in  the  divisor. 

2.  If  the  places  in  the  quotient  are  not  so  many  as  the 
rule  requires,  supply  the  defect  by  prefixing  cyphers  to 
the  left  hand  of  said  quotient. 

NOTE.  If  the  decimal  places  in  the  divisor  be  mere 
than  those  in  the  dividend,  annex  as  manv  cyphers  to  the 
dividend  as  you  please,  so  as  to  make  it  equal,  (at  least) 
to  tho  divisor.  Or,  if  there  be  a  remainder,  you  may 
annex  cyphers  to  it,  and  carry  on  the  quotient  to  any  de- 
gree of  exactness. 


EXAMPLES. 


9,51)77,4114(8,14  S,8),21318(,0561 

76,08  190 

231 

228 


3804  38 

3804  38 

00  00 

3.  Divide  780,517  by  24,3            Answers.  32,12 

4.  Divide  4,18  by  ,1812  ,23068  -f 

5.  Divide  7,25406  by  957  .00758 

6.  Divide  ,00078759  by  ,525  ,00150-f 

7.  Divide  14  by  365    '  ,058356+ 

8.  Divide  8246,1476  by  g604,25  ,40736-f 

9.  Divide  g!8651S,239  by  gS04,81  6ll,9-f 

10.  Divide  gl,28  by  g8.3l  ,154-f- 

11.  Divide  56cts.  by  1  dol.  12cts.  ,5 
1£.  Divide  1  dollar  by  12  cents.                          8,335 -f 
13.  If  21^  or  21,75  yards  of  cloth  cost  54,317  dollars, 

what  will  one  yard  cost  ?  SI, 577 

NOTE.     When  decimate,  or  whole  numbers,  are  to  be 
divided  by  10,  100, 1000,  &c.  (viz,,  unity  with  cyphers) 


f  DECIMAL    FRACTIONS.  83 

, 

it  is  performed  by  removing  the  separatrix  in  the  divi- 
dend, so  many  places  towards  the  left  hand  as  there  ave 
cyphers  in  the  divisor. 


EXAMPLES. 


{10,  the  quotient,  is  57,2 
100,  -  -  5,72 
1000,  -  -  ,572 


REDUCTION  OF  DECIMALS. 
CASE  I. 

To  reduce  a  Vulgar  Fraction  to  its  equivalent  Decimal. 
RULE. 

Annex  cyphers  to  the  numerator,  and  divide  by  the 
denominator ;  and  the  quotient  will  be  the  decimal  re- 
quired. 

'4  NOTE.  So  many  cyphers  as  you  annex  to  the  given 
numerator,  so  many  places  must  be  pointed  in  the  quo- 
tient; aiui  it  there  be  not  so  many  places  of  figures  in 
the  quotient,  make  up  the  deficiency  by  placing  cyphers 
to  the  left  hand  of  the  said  quotient. 

EXAMPLES. 

1.  Reduce  -J  to  a  decimal.  8)1,000 

Jns.     ,125 

2.  What  decimal  is  equal  to  {.  ?        Answers.  ,5 
5.  What  decimal  is  equal  to    £  ?       -     -     -     -      ,75 

4.  Reduce  ^  to  a  decimal.      ------         ,£ 

5.  Reduce  -]•£   to  a  decimal.      -----        ,6875 

f>.  Reduce  I ;J  to  a  decimal. ,85 

7    Jl'.-ijig^  toa  decimal. ,09375 

8.  What  decimal  is  equal  to  ^I       -     -     ,037057+ 

9.  Reduce  y  to  a  decimal.       -     -     -     -     ...33333  + 

10.  Reduce  17\T  to  its  equivalent  decimal.      -    ,008 

11.  Reduce  &  to  a  decimal.      -    -     -        ,1923076+ 


84 


DECIMAL  TRACTION'. 


SE    II. 

To  reduce  quantities  of  several  denominations  to  a 
Decimal. 

RULE- 

Bring  the  given  denominations  first  to  a  vulgar  fraction 
by  Problem  TIL  page  76 ;  and  reduce  said  vulgar  frac- 
tion to  its  equivalent  decimal ;  or 

RULE  2.  Place  the  several  denominations  above  each 
other,  letting  the  highest  denomination  stand  at  the  bot- 
tom ;  then  divide  each  denomination  (beginning  at  the 
top)  by  its  value  in  the  next  denomination,  the  last  quo- 
tient will  give  the  decimal  required. 

EXAMPLES. 

1.  Reduce  12s.  6d.  Sqrs.  to  the  decimal  of  a  pound. 
12 

150 
4 

960)603,000000(,628125    Answer. 
5760 


2700 
1920 

7800 
7680 

1£00 
960 

2400 
1920 


By  Rule  5. 


6,75 
12,5625 


,628125 
6 


4800 
4800 


DECIMAL  fRAcrioNS. 


'J2.  Reduce  15s.  9d.  3qrs.  to  the  decimal  of  a  pound. 

.Ins.  ,790625 
S.  Reduce  9d.  Sqrs.  to  the  decimal  of  a  shilling. 

Ms.  ,8125 

4.  Reduce  3  farthings  to  the'decimal  of  a  shilling. 

Ms.  .0625 

5.  Reduce.  3s.  4d.  New-England  Currency,  to  the  dc 
eimalof  a  dollar.  Ms.  ,555555 -f- 

6.  Reduce  12s.  to  the  decimal  of  a  pound,    .ins.  .0 

NOTE.  When  the  shillings  are  even,  half  the  numbci 
with  a  >oim  prefixed,  is  their  decimal  expression ;  but 
if  tlie  number  be  odd,  annex  a  cypher  to  the  shillings,  and 
then  by  halving  them,  you  will  have  their  decimal  ex- 
pression. 

7.  Reduce  1,  2,  4,  9,  16  and  19  shillings  to  decimals 
Shillings     1         2        4        9         16         19 

Msu'ers.  .05        ,1       .2      ,45         ,8       ,95 

8.  What  is  the  decimal  expression  of  4/.  19s.  G.Jd.  ? 

Ms.  £4,97708-1 

9.  Bring   S4£.  16s.  7$d.  into  a  decimal  expression. 

Ms.  £34,83229 1G-J- 

10.  Reduce  2,51.  19s.  5Jd.  to  a  decimal. 

Ms.  £  25,9729  JG-f 

11.  Reduce  Sqrs.  2na.  to  tlie  decimal  of  a  yard. 

Ms.  ,875 

12.  Reduce  1  gallon  to  the  decimal  of  a  hogshead. 

Mi.  .015873^;. 

13.  Reduce  7oz.  19pwt.  to  the  decimal  of  alb.  tn>y. 

Jin s.  .(562.5 

14.  Reduce  Sqrs.  Sllb.  Avoirdupois, to  the  doc'nvnl  c.f 
an  owt.  .-ins.  ,937.5 

15.  Reduce  2  roods,  1G  perches  to  tlie  decimal  of  an 
acre.  Jin' 

16.  I'cJuce  2  feet  G  inches  to  the  decimal  of  a  ; 

irfvio 

juris,  ,>>o.-o.>o- j- 

17.  Reduce  ofur.  iGpo.  totlic  dcciiKal  o!  a  • 

IS.  Reduce  4^  calendar  months  to  the  d 

8. 


86  DECIMAL    FRACTIONS. 

CASE  III. 

To  find  the  value  of  a  decimal  in  the  known  parts  of  th» 
integer. 

RULE. 

1.  Multiply  the  decimal  by  the  number  of  parts  in  the 
next  less  denomination,  and  cut  oft' so  many  places  for  a 
remainder,  to  the  right  hand,  as  there  are  places  in  the 
given  decimal. 

2.  Multiply  the  remainder  by  the  next  inferior  denom- 
ination, and  cut  off'  a  remainder  as  before ;  and  so  on 
through  all  the  parts  of  the  integer,  and  the  several  de- 
nominations standing  on  the  left  hand,  make  the  answer 

EXAMPLES. 

1.  What  is  the  value  of  ,5724  of  a  pound  sterling? 
£•  ,5724 
20 


11,4480 
12 


1,5040  Jlns.  11s.  5d.  l,5qr». 

2.  What  is  the  value  of  ,75  of  a  pound  ?     J)ns.  15*. 

3.  What  is  the  value  of  ,85251  of  a  pound  ? 

Ans.  17s.  Od.  2,4gra. 

4.  What  is  the  value  of  ,040625  of  a  pound  ? 

Jlns. 

5.  Find  the  value  of  ,8125  of  a  shilling.      Jlns. 

6.  What  is  the  value  of  ,617  of  an  cwt. 

J/i.s.  2</rs.   1Mb.  los.  10,6Jr. 

7.  Find  the  value  of  ,76442  of  n  pound  troy. 

Jlns.  9o*.  Spiet.  ll#r. 

8.  What  is  the  value  of  ,875  of  a  yd.  ?   Jlns.  Sqrs.  2« 

9.  What  is  the  value  of  ,875  of  a  hh«l.  of  wine  ? 

Jlns.  55 gal.  Off.  lj»f. 


8? 

t(j.  Fi-i-i  the  pi  v>f  a  mile. 

.flns.  28  !//..  11,  Win. 

11.  .  '..T  quantity  of  ,90  ."5  of  an  acre. 

J//s.  Sr.  25,l2j,'/>. 

12.  What  is  the  value  of  ,569  of  aye;ir  of  565  days-  ? 


13.  What  is  the  proper  quantity  of  ,002084  of  a  pound 
troy  :  Ms.  12,00384  -r. 

14.  What  is  the  value  of  ,046875  of  a  pound  avoirdu- 
pois P  Jlns.  I2,dr. 

15.  A  V  hat  is  the  value  of  ,712  of  a  furlong? 


16.  Wnatis  the  proper  quantity  of  .142465  of  a  year  r 


CONTRACTIONS  IN  DECIMALS. 
PROBLEM  I. 

A  CONCISE  and  easy  method  to  find  the  decimal  of 
any  number  of  shillings,  pence  and  fax-things,  (to  three 
places)  by  INSPKCTIOX. 

RULE 

1.  Write  half  the  greatest  even  number  of  shillings  for 
the  first  decimal  figure. 

2.  Let  the  farthings  in  the  given  pence  and  farthings 
possess  the  second  and  third  places ;  observing  to  increase 
the  second  place  or  place  of  nundredths,  by  5  i!'  the  shil- 
lings be  odd;  and  the  third  place  by  1  "when  the  far- 
tilings  exceed  12,  and  by  2  when  they"  exceed  36. 

EXAMPLES. 

1.  Find  the  decimal  of  7s.  9$d.  by  inspection. 
,3     =*  6s. 

5      for  the  odd  shillings. 

39=the  farthings  iii  9jd. 

2    for  the  excess  of  36. 

£.  ,391  =dccimal  required. 


DECIMAL   FRACTIONS. 

2.  Find  the  decimal  expression  of  16s.  4$d.  and  17 
8id.  Jus.  £.  .819,  onrf  £.  ,885 

3.  Write  down  £47  18  10 J  in  a  decimal  expression. ! 

dns.  £47,945 

4.  Reduce  £1  Ss.  2d.  to  an  equivalent  decimal. 

Ms.  £1,408 

PROBLEM  II. 

A  short  and  easy  method  to  find  th^  value  of  any  deci- 
mal of  a  pound  by  inspection. 

RULE. 

Double  the  first  figure,  or  place  of  tenths,  for  shillings, 
and  if  the  second  figure  be  5,  or  more  than  5,  reckon 
another  shilling;  then,  after  this  5  is  deducted,  call  the 
figures  in  the  second  and  third  places  so  many  farthings, 
abating  1  when  they  are  above  1 2,  and  2  when  above  36, 
and  the  result  will  be  the  answer. 

NOTE.  When  the  decimal  has  but  2  figures,  if  any 
tiling  remains  after  the  shillings  are  taken  out,  a  cypher 
roust  be  annexed  to  the  left  hand,  or  supposed  to  be  so. 

EXAMPLES. 

1.  Find  the  value  of  £.  ,i79,  by  inspection. 

12s.t=doublcot  6' 
1       for  the  5  in  the  second  place  which  is  to 

[be  deducted  out  of  7. 

Add  7$d.=29  farthings  remain  to  be  added. 

Deduct          id.  for  the  excess  of  12. 

Ms.  13s.  7d. 

2.  Find  the  value  of  £.  ,876  by  inspection. 

Jw.«.   17s.  6irf. 

5.  Find  the  value  of  £.  ,842  by  inspection. 

Jus.  16s.  lOrf. 
4.  Find  Hie  value  of  £t  ,097  by  inspection. 


OF   CURRENCIES  89 

:iEI)UCTK)N  OF  CURRENCIES. 

RULES, 

I1  OR  reducing  the  Currencies  of  the  several   United 
States*  into  Federal  Money. 

CASE  L 

To  reduce  the  currencies  of  the  different  states,  where 
a  dollar  is  an  en:  n  number  of  shillings,  to  Federal  Money. 
They  are 

f~ New -England^  JVfcw-  Fort,  and  ) 

j  Virginia',  North-Carolina,  $ 

^j  Jti'iitucky,  and 
\JKnnessee. 

RULE. 

1.  \Vhen  the  sum  consists  of  pounds  only,  annex  a  cy- 
pher to  the  pounds,  and   divide  by  half  the  number  of 
shillings  in  a  dollar  ;  the  quotient  will  be  dollars. f 

2.  But  if  the  sum  consists  of  pounds,  shillings,  pence, 
&c.  bring  the  given  sum  into  shillings,  and  reduce  the 
pence  and  farthings  to  a  decimal  of  a  shilling  ;  annex  said 
decimal  to  the  shillings,  with  a  decimal  point  between,  then 
divide  the  whole  by  the  number  of  shillings  contained  in 
a  dollar,  and  the  quotient  will  be  dollars,  cents,  mills,:&c. 

*  Formerly  the  pound  w;is  of  the  same  sterling  value  in  all 
tlie  colonies  as  in  Great-Britain,  arid  a  Spanish  Dollar  worth 
4sG — but  the  legislatures  of  the  different  colonies  emitted  bills 
of  credit,  which  afterwards  depreciated  in  their  value,  in 
some  states  more,  in  others  less,  &c. 

Thus  a  dollar  is  reckoned  in 


JWu»-  En  gland, ,""] 
Virginia,  ' 

J&ntucky,  and 


J 

,A'.  Carolina."  > 


Jfete-Jersey, 

Pennsylvania,  1  ^  <, 
Delaware;  and 

Maryland. 


South* 

Carolina 


-I 

Sia.  j 


4s8 


a  cypher  to  the  pounds,  multiplies  the  whole  by 
10,  bringing  them  into  tenths  of  a  pound  ;  then  because  a 
dollar  is  just  three-tenths  of  a  pountl  N.  E.  currency,  divi- 
fVin£  those  tenths  by  3,  brings  them  into  dollars,  kr.  See 
Note,  pa^e  85.  * 

8  » 


REDUCTION   OF   CURRENCIES. 

EXAMPLES. 

).  Reduce  737.  New-England  and  Virginia  t'flrrencj, 
:».  Federal  Monev.  5)730 

g  cts. 

g24S->=243  33^ 

2.  Reduce  45l.  15s.  7 Id.  New-England  currency,  to 

20  ^federal  money. 

AdoUar=G)915,623  12)7,500 

£152,604+     ^ns.  5625   decimal. 

NOTK.  1  farthing  is  ,251  which  annex  to  the  pence, 

2  —     =    ,50  i. and  divide  by  12,  you  wfll 

3  —     =    ,75  J  liave  the  decimal  required. 

3.  Reduce  345/.  10s.  lljrf.  New-Hampshire,  &c.  cur 
rencv,  to  Spanish  milled  dollars,  or  federal  money. 

£345     10     11* 

20  d .  ' 

12)11,2500 


6)6910,9375 


J9271*  decimai. 


SI  151, 8229+  .Ins.  ' 

4.  Reduce  105/.  14s.  S|rf.  New-York  and  North-Caro- 
lina currency,  to  federal  money. 

£105     14    3$  d. 

20  12)3,7500 

A  dol!ar=8)21 14,3125  ,3125  decimal. 

$264,289  06  Jns.  , 
Or  g  rfc7n.    o  V 

5.  R<Mluce  43 1 1.  New-York  cursency  to  federal  money. 
This  being  pounds  only.* —    4)4310 

g  eft.  • 
Ans.  g!077$=1077,50  ' 

*.i  diillar  is  &>.  in  this  currency— -,4=  -*s  of  a  pound  ; 
Iherrfurt*  multiply  by  10,  and  divide  by  4,  brings  ike 
puunds  intu  dollars, 


REDUCTION  OP   OURHEMOIES.  91. 

C.  Reduce  28Z.  11s.  6d.  New-England  and  Virginia 
currency,  to  federal  money.  Jins.  $95,  25c£s. 

7.  Change  46S/.  10s.  8d.  New-England,  Sec.  currency, 
-to  federal  money.       .  Jins.  81545,  llcfc.  Iw.-f- 

8.  Reduce  351.  19s.  Virginia,  &c.  cun-eiiry,  lo  ilideral 
money.  Jins.  §119,  BScfs.  3?n.-|- 

9.  Reduce  21  4i.  10s.  7£d.  New-York,  &c.  currency, 
to  federal  money.  Jins.  S536,  32rte.  8m.  + 

10.  Reduce  304/.  11s.  5d.  North  -Carolina,  &c.  cur- 
rency, to  federal  money.        Jins.  g76l,  42c£s.  7w.+ 

11.  Change  219i.  Us.  7|d.  New-England  and    Vir- 
ginia currency,  to  federal  money.      Jins.  g7Sl,  9-.r^.+ 

12.  Change  241/.  New-England,  &.c.  currency,  into 
federal  money.  Jins.  g803,  33cfs.+ 

15.  Bring  20/.  18s.  5£d.  New-England  currency,  into 
dollars.  ,  Jins.  S69,  74cts.  GA?».+ 

14.  Reduce  468  Z.  New-York  currency,  to  federal  mo- 
ney. Jins.  81170 

15.  Rfyiuce    17s.  9^d.  New-York,  &c.   currency,  to 
dollars,  &c.  J?is.  §2,  22cfs.  6,5>;i.-f 

16.  Borrowed  10  English  crowns,  at  6s.  3d.  each,  how 
many  dollars  at  6s.  ea£n,  will  pay  the  debt  ? 

Jins.  gll,  llcfs.  1m.  wt 

NOTE.  —  There  are  several  short  practical  methods  ^ 
reducing  New-England   and   New-York   currencies 
Federal  Money,  for  which  see  the  Appendix. 

CASE  II. 

To  reduce  the  currency  of  New-Jersey,  PennsyJ 
Delaware  and  Maryland,  to  Federal  Mbnc  ' 
RULE. 


Multiply  the  given  sum^by  8,  and  divide 
j^anu  the  quotient,  wi^l  be  dollars,  ^cc. 


ie 


,-?  ^0  federal 
1.  Reduce  245J.  New-Jersey,  &c.  curren"" 

7noncJ-  ;.'S3ic/s. 

£245x8=1960,  and   1960-7-3  =gG5:U=g^c>  in   the 

NOTE.  —  When  there  are  shillings,  pence,  _: 

-  --  ,  -  . 

*J1  dollar  is  7s.  6</.=90J.  in  this  currencuts 
tt  pound  ;  therefore,  vwltiplying  by  8,  and  divit 
Drives  the  dollars,  cents,  £(c. 


92  DEDUCTION    OF    CURRENCIES. 

given  sum,  reduce  them  'to  the  decimal  of  a  pound,  then 
multiply  and  divide  as  above,  &c. 

2.  Reduce  36J.  11s.  8}d.  New-Jersey,  &c.  currency, 
to  federal  money.    £36,5854  decimal  value. 

8 

8 

3)292.6832(97,56106  Ans.     ANSWERS. 
£.      s.    d.  g     cts.  m. 

3.  Reduce  240    0    0  to  federal  money  640  00 

4.  Reduce  125     8    0         334  40 

5.  Reduce     99    7    6*      265  00  5    -f 

6.  Reduce  100    0    0        266  66  6    -f 

7.  Reduce     2J    3    7  67  14  4 

8.  Reduce      0  17    9        2  36  6,6 

CASE  III. 

To  reduce  the  currency  of  South-Carolina  and  Georgia, 
to  federal  money. 

RULE. 

Multiply  the  given  sum  by  30,  and  divide  the  product 
by  7,  the  quotient  will  be  the  dollars,  cents,  &c.* 

EXAMPLES. 

1.  Reduce  100/.  South-Carolina  and  Georgia  currency, 
u  '••deral  money. 

100J.xSO=SOOO ;  SOOO-r-7=g42S,5714  Jns. 
m,me'ieduce  541.  16s.  9jd.  Georgia  currency,  to  federal 
54,8406  decimal  expression. 
SO 


'  '.645,2180 

jJns.  - 

£35,0311  *                 ANSWERS. 

S   Reduce  £•      s>    <*•  *                       8    cts.  in. 

4   Reduce  ^4    14    8  to  federal  money,  405   99    84- 

5!  Reduc^    19    17>    6^  -                85    18    7~ 

C.  Rediv*  417    14    6  -            179°   ^ 


°e  14°     10    °          -  602    14    2 

,-Ar.B  160      00  685   71    4 


-  or  56rf.  <o  the  d»War=/j6TaeI7ir  of  «  pound; 
fure  xSO-r-7. 


C  iiiON'    Of 


.<:.      (/. 

8.  i  0     li.     6  -  4-f- 

9.  Reduce  41     iT    9  -  J7y  ol  4A 

CASE    IV. 

To  reduce  the  currency  of  Canada  and  Nova-Scotia,  to 
Federal  Money. 

RULE. 

Multiply  the  given  sum  by  4,  the  product  will  he  <!• 
NOTE.  Five  shillings  of  this  vqiMi  to  a 

dollar;  consequently  4  dollars  ma^e  ouop, 


I.  Reduce  125?.  Canada  and  Nova-Sec  -•}%  to 

federal  money.  125 

4 


s.  S500 

2.  Reduce  55l.  10s.  6rf.  Nova-Scotia  currency,  to  dol- 
lars. 55,525  decimal  value. 

4 

-  -      S   cts. 
Ms.  S222,  100=22-}  10  ANSWERS. 

3.  Reduce  241  18     9  to  federal  money.  967  75 

4.  Reduce     58  13     6}  -  *  234  70 

5.  Reduce  528  17     8  --  2115  5.1 

6.  Reduce       120  -  4  50 

7.  Reduce  224  19    0  9  80 

8.  Reduce      0  13  Hi  - 


REDUCTION  OF  COIN. 
RULES 

For  reducing  the  Federal  Money  to  the  cum 

several  United  States. 
To  reduce  Federal  Money  to  the  currency  of 


w-Englaad^        rMuUSply:-tilp  sivcn  cnm  i,y  ,G 
jWS     ,nd  >2  \  ^d  the  product  will  bcF 
' 


decimals  of  a 


94  REDUCTION    OF     COIN. 

LV.       .vl  *  Iilu,hiply  lh,e  ?ve.v,  rum  by/ 

-'•  1  A*.  Carolina:  (  3  ]  ^  the  product  will  be  pounds, 
J  p;  (^and  decimals  ot  a  pound. 


.Wir-Jpr-vv,  "^1  .    f  Multiply    the  given  sum  by    3 

\  Pennsylvania,  !  ^  J  and  divide  the  product  by  8,  & 

'\  Delaware,  8{    j  ^   j  the  (juotient  will  be   pounds, 

\jMarijland.  "   [^atnl  decimals  of  a  pound. 


3    J 


ft,     .,  /n,      ,.      "1        fMultiplv  the<rivensumby,? 
[South-Carolina,  U^M^^jS  ,thequctient 
.  I   =M  will  l>e  the  ansver  in  pounds. 

Georgb.         J  P:   Laud  decimals  of  a  po^d. 


EXAMPLES, 
In  the  foregoing  Rules. 

1.  lieduce  gl52,  GO  cts.  to  New-England  currency. 
3 


£45,  780  Jlns.=£45  15s.  7,2rf. 

^0       But  tlie  value  of  auy  decimal  of 

a  pound,  may  be  found  by  inspeo 

15,  600  tion.     8ec  problem  II.  page  88. 


7,  200 

2.  In  $196,  how  many  pounds,  N.  England  currency. 
S 

£58,8  »flrts.=£58  16 

r>.  Reduce  £629  into  No  \v-York,  &C.  currency. 
,4 


!.t>  .7">-.=/;-i'i  12 

4.  Bring   S110,  5i   cts.^i   in.  into  New-Jersey,  &c. 

t  UITU11CV. 


REDUCTION    OF    COIN.  95 

8110,511 
S 

jJouble  4  makes  8s.     Then   59  far- 

8)331,533     tilings   is    9d.    Sqra.     See  Problem   II. 

—    page  88. 

£41,441  »fl?js.=£41  8s.  9|fZ.  bu  Inspection. 
5.  Bring   £65,  36  cts.  into  South-Carolina,  &c.  cuu- 
rency.  ,7 

3),45,  752 


5s.  «3«s^  ANSWBRS. 

S  cfs.  £.     S.    d. 

6.  Reduce  425,07  to  N.  E.  &c.  currency.  127  10    5  + 

7.  Reduce     36,1 1  to  N.  Y.  &c.  currency.  14     810*4- 

8.  Reduce  315,44  to  N.  J.  &c.  currency.  118     5     9i+ 

9.  Reduce  690,45  to  S.  C  &c.  currencyt  161     2   1,2 


To  reduce  Federal  Money  to  Canuaa  and  Nova-Scotia 
Currency. 

RULE'. 

Divide  the  Dollars,  &c.  by  4,  the   quotient  wm  ue. 
pounds,  and  decimals  ot  a  pound. 

EXAMPLES. 

1.  Reduce  g741  into   Canada  ami  Nova-Scotia  cur- 
rency, g  cts. 
4)741,00 


£185,25 =£185  5s. 
2.  Bring  $311,  75  cts.  into  Nova-Scotia  currency. 

g  cts. 
4)311,750  ' 


£77,9375=£77  18s.  9d. 

3.  Bring  g290~,  56  cts.  into  Nova-Scotia  currency. 

Jhif.  £726  17s.  9id. 

4.  Reduce  S2114,  50  cts.  into  Canada  currency. 

Ans.  £528  l£s.  fi-L 


>   FOR   REDUOIXO,   OU 


HU'  ing  the  currem  ial  I'm- 

•-•  others.  «  currency 

od  currency, 


• 

.  land, 
inia, 
ttcfcy, 

and 

Trn-tesffe. 

Dtlawure, 
and 
Maryland. 

J>V;r-I"orfc, 
«7?J 
cA*.  Carolina. 

'rlaitd, 
Kentucky, 

Ten), 

Add  one  4tli 
given 
sura. 

Add  one  3d 
to  the  given 
sum. 

'AV/r-. 
Penn?y-        < 
Delaware. 

ail  «. 

q,'.ven  ? 

Add  one  5f- 
tn-ntls  to  the 
-urn.  • 

••     '  - 
Ac 

Deduct  one 

'r'.i  ft 
'<  »rk. 

n.-duct  one 
itn  tlio 
New-York. 

| 

."id. 

-|yt!:o 

;  rodnct 
in-  7. 

Multij.ly  the 
^ivi-n  .Miin  by 
divide 
tiio  product 
by  2S. 

Mulfij.ly  the 
ur-i'ii  sum  hv 
•i.i    di- 
c  pro- 
duct bv  7. 

1 

Ai'-l  onc5t!i 

c. 

___ 

Add  nnchuil' 

;  ly  the 

• 
divide 

the     ji 

•  :d  di- 
vidv  the   pru- 

' 

• 

duct  !> 

FOR    KKDUCIN'O,   &C.  97 

ted  States,  also  Canada,  Nova-Scotia,  and  Sterling,  eacji 
in  the  left  hand  column,  and  then  cast  your  eye  to  thfc 
and  you  will  have  the  rule. 


South-Caroliiui, 
and 
Georgia, 

Canadj, 
and 
Nova-Scotia. 

Sterling. 

Multiply  the  giv- 
en sum  by  7,  and 
divide  the  product 
by  9. 

Multiply  the  giv- 
en sum  by  5,  ami 
divide  the  product 
by  6. 

Deduct  one 
fourth  from  the 
given  sum. 

Multiply  the  giv- 
en sum  by  28,  and 
divide  the  product 
by  45. 

Deduct  one  third 
from  the  given 
sum. 

Multiply  the  giv- 
en sum  by  5,  and 
divide  the  product 
by  5. 

Multiply  the  giv- 
en sum  by  7,  and 
divide  the  product 
by  12. 

Multiply  the  giv- 
en sum  by  5,  and 
divide  the  product 
by  3. 

Multiply  the  giv- 
en sum  by  0,  and 
divide  the  product 
by  16. 

Multiply  the  giv- 
en stun  by  15,  and 
divide  the  product 
by   14. 

From  the  given 
sum,   dedu. 
twenty-eighth. 

Deduct  one  fif- 
teenth from  the 
given  sum. 

Deduct  one 
ienth  tVom  the 
given  sum. 

Add  one  ninth 
to  the  given  sum. 

X>  the  English 
mutiny  add  one 
tvvccty  -seventh. 

*  REDUCTION    OF    COIN. 

APPLICATION 

Of  the  Rules  contained  in  the  foregoing  Table. 
EXAMPLES. 

1.  Reduce  461.  10$.  &/.  of  the  currency  of  New-Hamjj; 
shire,  into  that  of  New-Jersey,  Pennsylvania,  &c. 

£.  s.  d. 

See  the  Rule  4)46  10  6 

in  the  Table.  -f  11  12  7* 


Jns.  £58     3  H 

2.  Reduce   25Z.    135.   9<f.  Connecticut    currency,  to 
New-York  currency. 

£.  5.  d. 

3)25  IS  9 

By  the  Tuble,+J  &c.  +8  11  3 

Jlns.  £34    50 

3.  Reduce  12JJ.  10s.  4d.  New-York,  &c.  currency,  to 
South-Carolina  currency. 

£.  s.  d. 

Rule  by  the  Table,  125  10  4 

xTV-r-by  12,  &e.  7 

12)878  12  4 


£73     4  4} 

4.  Reduce  -4G/.  1  15.  8rf.  New-York  and  North-Caro- 
irrcncy,  to  sterling  or  English  Monev. 

£.  5.  "d. 

46  11  8 

9 


Sfeii  •]  lG.=4x4N.419  5  0 
•n  sum  by  v  4)104  16  3 
.&c.  j  .  r^- 


Ar.s.  £26    4  Oj 


REDUCTION    OF     COIN. 

To  reduce  any  of  th  different  currencies  of  the  sevb 
ral  States  into  each  other,  at  par ;  you  may  consult  the 
preceding  Table,  which  will  give  you  the  Rules. 

MORE    EXAMPLES   FOR   EXERCISE. 

5.  Reduce  841.  10s.  8d.  New-Hampshire,  &c.  curren- 
cy, into  New  -Jersey  currency. 

Jlns.  £105  15s.  4d. 

6.  Reduce   ISO/.  8s.  3d.   Connecticut  currency,  into 
New-York  currency.  Jlns.  £160  lls.  Gd. 

7.  Reduce    ICO/.    10s.   Massachusetts  currency,  into 
South-Carolina  and  Georgia  turre  icy. 

*  "  Jlns.  £93  14s.  5i<7. 

8.  Reduce  410i.  1F>\  lid.  "Rhode-Island  currency,  in* 
to  Canada  and  >  ova -Scotia  currency. 

>,zs.  £342  9s.  Id. 

9.  Reduce  5241.  8s.  4d.  Virginia,  &c.  currency,  into 
Sterling  money.  Jlns.  £393  6s.  3d. 

10.  Reduce  214Z.  9s.  2d.  New-Jersey,  &c.  currency, 
into  New -Hampshire,  Massachusetts,  &c.  currency. 

.flns.  £171  lls.  4d. 

11.  Reduce  100Z.  New-Jei-sey,  &c.  currency,  into  N. 
York  and  North-Carolina  currency. 

.JJns.  £106  13s.  4d. 

12.  Reduce  WOl.  Delaware  and  Maryland  currency, 
into  Sterling  money.  Jlns.  £60. 

13.  Reduce  1162.  10s.  New-York  currency,  into  Con- 
necticut currency.  w2ns.  £87.  7s.  6d. 

14.  Reduce   llQl.   7s.   Sd.  S.  Carolina  and  Georgia 
currency,  into  Connecticut,  &c.  currency. 

Jns.  £144  9s.  3  Id. 

15.  Reduce  100Z.  Canada  and  Nova-Scotia  currency, 
into  Connecticut  currency.  Atis.  £  120. 

16.  Reduce  116J.  145.  9d.  Sterling  money,  into  Con- 
necticut currency.  Jlns.  £155  13s. 

17.  Reduce  104J.  10s.  Canada  and  Nova-Scotia  cur- 
rency, into  New-York  currency.  Jlns.  £167  4s. 

18.  Reduce  WQL  Nova-Scotia   currency,  into  New- 
Jersey,  &c.  currency.  Jlns.  £150 


RULE    OF    THREE     DIRECT. 

RULE  OF  THREE  DIRECT. 

rl* 
i  HE  Rule  of  Three  Direct  Teaches,  by  having  tliree 

numbers  given  to  find  a  fourth,  which  shall  have  the  same 
proportion  to  the  third,  as  the  second  has  to  the  first. 

1.  Observe  that  two  of  the  given  numbers  in  your 
question  are  always  of  the  ?;  le,  or  kind;  one  of 

which  must  be  the  first  number  in  stating,  and  the  other 
the  third  number  :  consequently,  the  first  and  third  num- 
bers must  alwa>>  ')e  of  the  same  name,  or  kind  :  and  the 
other  number,  which  is  of  the  same  kind  with  the  answer, 
or  thing  sought,  will  always  ;  ic  second  or  middle 

place. 

l-.  The  third  term  is  a  demand^and  may  be  known  by 
•  or  the  like  words  before  -t.  >  i/..  What  will ;  "What 
cost?  Kow  mair  ?  JIuw  for?  How  long?  or,  How 
much  ?  &c. 

1.  State  the  question:  ;  lice  the  numbers  s» 
that  the  first  and  i  ie  kind; 
and  the  second  term  of  the. same  kind  with  the  answer,  or 
thing  sought. 

2.  Bring  the  ir-t  and  third  terms  to  the  same  denom- 
ination, and  reduce  T;<O  second  term  to  the  lowest  name 
mentioned  in  it. 

3.  'Multiply  liie  second  and  third  terms  together,  and 
divide  their  product  by  i»>rm;  the  quotient  will 
be  the  an-  .:me  denomination 

,e  second  ten  .;iy  be  brought  into 

. 
T!  •  iiestion. 

icn  they 

'•rform  tl;  .:  a  much  shorter  mrmnar 

'.era I  rule. 

';ip!y  the.  quo- 
Or 

Or 

'1  oy 
Or 

first  term 
;j  the  last'     oi  the  answer,. 


RULE   OF   THREE    DIRECT.  101 

EXAMPLES. 

1 .  If  6  yards  of  cloth  cost  9  dollars,  what  will  20  yards 
cost  at  the  same  rate  ?  Yds.  g     Yds.  " 

Here  20  yards,  which  moves  6  :  9  :  :  20 

the  question,  is  the  third  term  :  9 
6yds.  the  same  kind,  is  the  first, 

and  9  dollars  the  second.  6)180 

Jlns.  gSO 

2.  If  20  yards  cost  SO  dols.       S.  If  9  dollars  will  buy  6 
what  cost  6  yards  ?  yards,  how  many  yards  will 

Yds.    8       Yds  30  dollars  buy  ?  , 

20  :  30  :  :  6    .  g    yds.       « 

6  9  :  6  :  :  30 

•  g 


2,0)18,0 
Ans.  &9 


9)180 


JJns.  ZQyds. 

4.  If  3  cwt.  of  sugar  cost  81.  8s.  what  will  1 1  cwt.  1  qr. 
24 Ib.  cost? 

3  cwt.  81.  8s.  C.  qr.  Ib.  Ib.         s. 

112  20         11  1     24    As    53G  :  168  :  :  1284M. 

4      r  168 

336  Ik.  168s.  — 
45 
28 


564 

92  556)215712(64,2 

2016     

1284/6  S2J.2s. 

1411    Jlns. 
1344 

672 
672 


»*ULK   OF    THREE  DIRECT- 


'ir  of  stockings  cost  -4s.  6d.  whnt  will  19 
-t  ? 

•iir  of  shoos  rest  51/.  6s.  what  \vili  one 
[fair  t  .  4s.  G</. 

r.  per  pound,  what  is  the  value,  ot"  a  firkin  of 

50  poum.  !  9s/ 

8.  •  h  sugar  can  you  buy  f<>:  'I.  a 

pound  ?  »5;/s.  5' 

Bmight  Schcsls  c.f  su^ar,  each  9  c\vi.  >•  hat 

»mo  to  at  2l.  5s.  per  cwt.  ? 

10.  If  a  man's  \vagcs  are  7C>[.  10s.  a  yti: 
atcahettdar 

11.  Jf  4i  tons  df  hay  will  kocp  S 

how  many  tons  will  it  take  to  keep  £5  cattle  tin*  same 

If  a  man's  yearly  income  1.  that 

V    ? 

.  If  a  man  spends  5s.  4d.  per  da--  .  hat 

14.  ,  at  iCs.  Gd.  per  wt      , 
10s.  last  me  ? 

15.  A  owes  B  3475^.  out  B  cnmp«n; 

>.  •  pound  j  pra 
his  (1. 

IT.  nith  sold  a  tankard  for 

\vliat  was  the  weight  (-i 

• 
. 


EXAMPLES. 


19.  If  7  yds.  of  cloth  cost  15  dollars  47  cents,  ^hst 
will  12  yds.  c<  Yds.    $cts.     yds. 

<       7  :  15,47  :  :'. 
12 

7)185,64 

JNS.  26,52  ==S2G,  52ete. 

Cut  any  sum  in  dollars  and  cents  may  beWritter 
as  a  whole  number,  and  expressed  in  its  lowest  den-, 
nation,  as  in  the  following  example:     (See Reduction 
Federal  Money,  page  67.) 

20.  What  will  1  qr.  9  Ib.  sugar  come  to,  at  6  dollars 
45  cts.  percwt.  ? 

qr.  Ib.  Ib.       cts.         Ib. 

1    9  As  112  :  645  :  :  37 

28  57 


37  Ib.  4515 

1935 

. cts 


112)23865(213+  .3ns.  =g2,  1. 

224 

146 
112 

345 


9 

NOTE  2.  When  the  first  and  third  numbers  are  fede- 

r;;l  money,  you  m-.iy  annex  cyphers,  (if  necessary)  unlil 

you   .  .-.Mires  at  the   right 

!  of  the  sep:ir;;trix,  •  .  e  them  t*> 

M  li  -  a.     Theii  yon  may  multiply  and  di- 

i-i.-l  tii^  (inotient  will  expres-! 
the  least  denomination  mentioned  iu  ths 


104  RULE    OF    THREE    DIRECT/ 

EXAMPLES^ 

21.  If  3  dolurs  \vi!I  buy  7  yards  of  cloth,  how  many 
yards  can  I  buy  for  12C  dollars,  75  cents? 
cts.     yds.       cts. 
As  300  ;  7  :  :  12075 
7 

---  yds. 
500)  84525(28  l}Mt. 
If  12  Ib.  of  Tea  cost  6  dols.          600 
78  cts.  and  r'  mills,  what  will  5  Ib.          - 
cost  at  the  same  rate  ?  2452 

Ib.     milis.         Ib.  2400 

As  12  :  6789  :  :  5  - 

5  525 

-  SOO 

12)53945  - 

gcfs.?JT.  225 

Ms   282a-f.jni//s,=r2,82,S.  4 

900(3yr5. 
900 

g    cts.  - 

23.  If  .1  man  lays  out  121,  23   in    merchandize,  and 
:')>  gains  r>0  dollars,  51  cts.  how  much  will  he  gain 
by  laying  out  12  dollars  at  the  same  rate  ? 

(H<s.       cents.       cents. 
As  12123  :  3951  :  :  1200 

1200 

-  cts.     g  cts. 
12123)4741200(391=5=3,91  Ans. 
36369 

110430 
109107 


1107 


•HUH    OF    THREE    DIRECT. 

24.  If  the  wages  of  15  weeks  com-e'to  G4  dols. 
what  is  a  year's  wages  at  that  rate  ? 

Jlns.  g222,  52cts.  5>>i. 

25.  A  man  bought  sheep  at  1  dol.  11  cts.  per  head,  to 
the  amount  of  51  dols.  6  cts.  j  ho\v  many  sheep  did  he 
buy  ?  Jlns.  46. 

26.  Bought  4  pieces  of  clot!),  each  piece  containing  31 
yards,  at  16s.  6d.  per  yard,  (New-England  currency) 
what  does  the  whole  amount  to  in  federal  money  ? 

Jlns 

27.  "When  a  tun  of  wine  cost  140  dollars,  v 
quart  ?  .,'//-.  13r.    , 

28.  A  merchant  agreed  with  his  debtor,  that  if  he 
would  pay  him  down  65  cents  on  a  dollar,  he  v.oulii 
him  up  a  note  of  hand  of  249  dollars,  88  cts.     I  demand 
what  the  debtor  must  pay  for  his  note  ? 

Jlns.  g!62.  4&-/S.  2?tt.' 

29.  If  12  horses  eat  up  30  bushels  of  oats  in  a  week, 
how  many  bushels  Avill  serve  45  horses  the  same  lit) 

Jin*. 

30.  Bought  a  pieca  of  cloth  for  g48  £7' 

19  cents  per  yard  j  how  many  yards  did  it  contain  ? 

"  Jin*.  40yds.  2  . 

31.  Bought  3  hhds.of  sugar,  each  weighintr  8  c\\; 
12 Ib. at  7  dollars,  26  cents percwt.  what  come  the, 

Jlns.  £. 

32.  What  is  the  price  of  4  pieces  of  cloth,  tlie  first 
piece  containing  21,  the  second  23,  the  third  24,  unu 
fourth  27  yards  at  1  dollar  43  cents  a  yard  ? 

Jlns.  £135  %5cis.  2l-j-23-f  24-J-27=95y(/.s. 

33.  Bought  3  hlids.  of  brandy,  containing  ("• 
gallons,  at  1  dollar,  38  cents  per  gallon,  1  denuuui 
much  they  amount  to  ? 

34.  Suppose  a  gentleman's  income  r,,  1830  dollars  a 
year,  and  he  spends  3  dollars  49  cents  a  day,  one 
•with  another,  now  much  will  he  have  sa 

«nd  ?  ..Jus.  j?.k 

35.  If  iny  horse  stands  me  in  20  cents  ptr 
ing,  what  will  be  the  charge  of  1 1  horses  for  th».- 
that  rate  ?  Jlns.  «: 


106  RULE    OF    THREE    DIRECT. 

Sf\  A  merchant  bought  14  pines  of  wine,  and  is  allow- 
ed 6  months  credit,  but  for  ready  money  gets  it  8  cents  a 
gallon  cheaper;  how  much  did  he  save  by  paying  ready 
money?  Ana.  gl41,"l2cenis. 

EXJIJIPLES— Promiscuously  placed. 

57.  Sold  a  ship  for  5371.  and  I  owned  |  of  her;  what 
was  my  part  of  the  money  ?  JHS.  £201  7s.  6d. 

38.  If  -,56  of  a  ship  cost  781  dollars  25  cents,  what  is 
the  whole  worth  r  S 

As  5  :  781,25  :  :  16  :  2500  Ans. 

39.  If  I  buv  54  yards  of  cloth  for  31f.  10s.  what  did 
it  cost  per  Ell  English  ?  Jin*.  14s.  7d. 

40.  Bought  of  Mr.  Grocer.  1 1  cwt,  3  qrs.  of  sugar,  at  8 
dollars  12  cents  per  cwt.  and  gave  him  James  Pavwell's 
note  for  19.'.  7s.  (New  >  urrci.cy)  the  rest  i  pay 
in  cash  :  tell  me   how   many  dollars    will  make   up  the 
balance  ?  Jins.  gSO,  91  ct*. 

41.  ll  as<aff  5  feet  long  casts  a  shade  on  level  ground 
8  feet,  what;  is  'Mat  steeple  whose  shade  at 
the  sit,  vet  ?  .  1 1 .-}'//. 

,,  gentle:''i..n  has  an  income  of  :"f);1  Englisli  gain- 

;ih  anoth- 
er, to  lay  up  500  d;;,  ie  yea.  s  r 

• .  82,  4f'C/s.  5m. 

43.  BoKght  50  pieces  of  kerseys,  each  34  Ells-Flemish, 
at  8i>.  4d.  per  Ell-English;  what  did  the  whole  • 

44.  Bo'-iaht  £0i-   v.inK  of  cambiirk  for  '."-0^.  hu(  beius; 
•:^ed,  lam  willing  to  iojp  71.  1()s.  by  •  f  it; 

.-t  I  demand  per  Ell-En 

45.  How  many  pieces  ot  Holla]  i-Kleiu- 
ish,  may  1  have  for  23/.  8s.  at  6s.  6d.  pi:r  E:!-l 

\  merchant  bought  a  l>a! 

;d  it 

it   1 1}  dollars  i--< 
by  the  bargain,  and  how 

St,is.  He  gained  fc 


RULE    OF    THREE  DIRECT.  107 

47.  Bought  a  pipe  of  wine  for  84  dollars,  and  found  it 
had  leaked  out  1&  gallons;  Isold  the  remainder  at  12J 
cents  a  pint;  what  did  I  gain  or  lose  ? 

»2ns.  I  gained  SSO. 

48.  A  gentleman  bought  18  pipes  of  wine  at  12s.  6d. 
(New-Jersey  currency)  per  gallon  ;  how  many  dollars 
will  pay  the  purchase  r  Jins.  gS780. 

49.  Bought  a  quantity  of  plate,  weighing  15  Ib.  11  oz. 
13p\vt.  17  gr.  how  many  dollars  will  pay  for  it,  at  the 
rate  of  12s.  7d.  New  -York  currency,  per  ounce  ? 

.•Ins.  gSOl,  50cfs.  2jV"- 

50.  A  factor  bought  va  certain  quantity  of  broadcloth 
and  drugget,  which  together  cost  81/.  the  quantity  ot 
broadcloth  was  50  yards,  at  1  8s.  per  yard,  and  for  every  5 
yards  of  broadcloth  he  had  9  yards  of  drugget;  I  demand 
now  many  yards  of  drugget  he  had,  and  what  it  cost  him 
per  yard  ?  .Ins.  90  yards  at  8s.  per  yard. 

51.  If  I  give  1  eagle,  2  dollars  8  dimes,  2  cents  and  5 
mills,  for  675  tops,  how  many  tops  will  19  mills  1 

. 
52*  Whereas  an  eagle  and  a  cent  just  three  scy:  e  yards 

did  buy, 
How  many  yards  of  that  same  cloth  for  15  dimes  ha:l  I  ': 

53.  If  the  Legislature  of  a  SMe  grant  a  tax  m' 

an  the  dollar,  how  much  must  that  man  pay  •.  <lo/l- 

lars,  75  cents  on  the  list  ? 

.#»*.  S^.  •~.T 

54.  If  100  dollars  gain  6  <1> 
how  much  will  49  dollars  gain  in  t 

55.  Jf  GO  gallons  of  water,  in  one  ho-,;. 
tern  con 

35  gallons  '-uu  out 
filled  ? 


iib.  A  ;>.r  d    R  Hryiart  fr- 
the  same  roail 

of  1 

76J. 


r  i  x 

El       . 

. 

. 

or  le.ss  >v;".'iT3  !•  ;es- 

Kule  of  Throe  Direct : 
requires  less  or  / 

reej  A  hkii 
•>\vii  from  the  nature  and  tenor  o! 

a'Mj'lt1  : 

5,  limv  many  day* 
v  ill  it  require  4  men  to  mo>\ 

7/u-ji 

require    4   how    rniuh    time    will      4     re- 
.•'.sv/er,  2  tlays.     Here  more  ir^ii!  \i/.. 

days 

re- 

- 
•I'UHie,  tl'.c  more  (L 

.  •  rse  Pioportion. 

RULE. 

1.  State  and  rediu  c  t';e  term?  as  in  the  Rule  < 
Dire 

nd 
divi, 

...  diMioirii-i.ititfu   as  tne   middle  teiiu 


1.  1»  . 

. 

NX  'M*   U-::{rl!l  Oi    boa!'. 
r.  «0  't  ^ 


10. 

4.  If  five  dollars  will  pay  for  the  carriage  of  2  cwt.  150 
miles,  how  far  may  15  cwt.  be  carried  for  the  same  mo- 
ney ?  Jinn.  20  miles. 

5.  If  when  wheat  is  7s.  6d.  the  bushel,  the  penny  loat 
will  weigh  9oz.  what  ought  it  to  weigh  when  wheat  is  6s,. 
per  bushel  ?  Jlns.  1 1  ox.  Cpwt. 

6.  If  30  J'ushels  of  grain,  at  50  eta.  per  bushel,  will 
pay  a  debt,  how  many  bushels  at  75  cents  per  bushel,  will 
pay  the  same  ?  Jlns.  20  bushels. 

7.  If  100J.  in  12  months  gain  61.  interest,  what  princi- 
pal will  gam  the  same  in  8  months  ?  Jlns.  £  150. 

8.  If  1 1  men  can  build  a  house  in  5  months,  by  work- 
ing 12  hours  per  day — in  what  time  will  the  same  num- 
ber of  men  do  it,  when  they  work  only  8  hours  per  day  ? 

Jlns.  7\  montfis. 

9.  What  number  of  men  must  be  employed  to  finish  in 
5  days,  what  15  men  would  be  20  days  about  ? 

Jlns,  60  men. 

10.  Suppose  650  men  are  in  a  garrison,  and  their  pro- 
visions calculated  to  last  but  two  months ;  how  many  men 
must  leave  the  garrison  that  the  same  provisions  may  be 
sufficient  for  those  who  remain  five  months  ? 

Jlns.  390  men. 

11.  A  regiment  of  soldiers  consisting  of  850  men  are 
to  be  clothed,  each  suit  to  contain  S$  yds.  of  cloth,  which 
is  1 2  yards  wide,  and  lined  with  shalloon  |  yard  wide; 
how  many  yards  of  shalloon  will  complete  the  lining  ? 

Jlns.  6941yds.  Zqrs.  Sfwa. 


PRACTICE. 

PRACTICE  is  a  contraction  of  the  Rule  of  'Hue? 
Direct,  when  the  first  term  happens  to  be  aa  unit  or  (.MI  , 
and  is  a  concise  method  of  resolving  n.opt  qacs 
occur  in  trade  or  business  when*  money  i.-=  reckoned  in 
Bounds,  shillings  and  pence:  but  reckoning  ir- 
Money  will  render  ij)i;. rule  al-nost  uncles*  :  t 
reason  !  shall  not  enlarge  so  much  on  the  subj^v. 
ry  ether  writers  have  dihie. 


110 


1WACTK11. 


Tables  of  Jlltquot,  or  Even  Tarts. 


Paris  of  a  Shilling. 
<L            s. 
6      is      4 

4      =•      ^ 
3               i 

2              * 

U 

Pails  of  2  Shillings. 
Is.     is     i 

Sd.    =    4 
C;l.             i 
4'.1.             1- 

&    A  . 

Parts  of  a  Pound. 

S.  d.            £. 
10  0   >     i 
6  8     «r.  j 

50            i 
40           £ 

34           j.- 
26 
1  8          TV 

-  uf  a  cwt. 
Ib.          cict. 
56     is    i 
28     =    i 
16          * 

The  aliquot  part  of  any  number, 
is  such  a  part  of  it.  as  being  taken  a 
certain   number   of  ^times,   exactly 
makes  that  number. 

CASE   I. 

the  price  of  one  yard,  pound.  &c.  is  an  even  part 
of  one  shilling. — Find  the  value  of  the  given  quantity  at 
Js.  a  yard,  pound,  &c.  and  divide  it  bv  that  oven  part 
;.!jrl  the  Quotient  will  be  the  answer  in  shillings,  &c. 

Or  find  the  value  of  the  given  quantity  at  2s.  per  yard, 
&o.  and  divide  said  value  by  the  even  part  which  the 
pvtn  price  is  of  £s.  and  the  quotient  will  be  the  answer 
in  shi  1  lings,  £c.  \\liichreducetopounds. 

N.  B.  To  find  the  value  of  any  quantity  at  2s.  you  need 
only  double  the  unit  figure  for  shillings;  the  other  fig- 
ures will  be  pounds. 

EXAMPI.KS. 

J.  What  will  461$  yards  of  tape  come  to.  at  l}d  ]  01 
J$d.  1 1  |  461  b  value  of  4d  1  ;  vd. 

5,7  8* 


£2  l?s.  8|J 

St.  WHiat  cost  256lh.  i.i I"  (  uul  ." 

8d.  |  ||  £25  12s.  value  ol 

£8  10s.  feJ.  jnd 


PRACU.Itf.  lit 

Tarn's,  per  y-'  •    £•  s-    &• 
4otU  at  lil.                    .Insurers,    206$ 

:     tit  2d.  738 

911     at  3d.  11     7    9 

749    at  4d  12    9     8 

113     at  6d  2  16    6 

891>    at  8d.  29  19    4 

CASE  II. 

When  the  price  is  an  even  part  of  a  pound — Find  the 
value  of  the  given  quantity  at  one  pound  per  yard,  £c. 
and  divide  it  by  that  even  part,  and  the  quotient  will  be 
the  answer  in  pounds. 

EXAMPLES. 

Wfcat  will  129$  yards  cost  at  2s.  6d.  per  yard  ? 
s.d.  £.  s.  £. 

2  G  |  j  |  129  10  value  at  1  per  yard. 


s.  £16  3s.  9ef.  value  at  2s.  6d.  per  yard. 

Yds.        s.   d.  £.   s.  rf. 

123    at  10  0  per  yard.  Answers.    61  10  0 

C87i  at    5  0      —  "  171  17  6 

21U  at     4  0      —  42     5  0 

543     at     6  8'      —  181     0  0 

127     at     3  4      —  21     3  4 

461    at    1  3      — i.  38    8  4 

NOTE.  When  the  price  is  pounds  only,4he  given  quan- 
tity multiplied  thereby,  will  be  the  answer. 

EXAMPLE. — 11  tons  of  hay  at  4l.  per  ton.    Thus    11 

'4 

Ans.  £44 
CASE  III. 

When  th«  given  price  is  any  number  of  shillings  u» 
<ler  20. 
1.  When  the  shillings  are  an  even  number,  multiply 


PRACTICE. 


jiiantityby  half  the  number  of  shillings,  and  doubl* 
--et  figure  of  the  product  for  sliillingsj  and  the  rest 
,?  product  will  be  pounds. 

the  shillings  be  odd,  multiply  the  "quantity  by  the 
\vhole  number  of  sliillin^a,  and  the  product  will  be  the 
:  in  shillings,  which  reduce  to  pounds. 


EXAMPLES. 


st.  124  yd*  at  8s 
4" 


£49 


Yds. 

*  4s. 

•  2s. 
913  ;:t  14s. 


Ss. 

132 
7 

yds.  at 

37 

659 

s. 

o 

16 

2 

2,0)92,4 

£46,4  dns. 

t£. 

572  at  11s.      A 
264  at     93. 
250  at  16s. 

per  yd. 


C.     s. 
dns.  204  1-2 

118  16 

200  00 


CASE  IV. 


When  flic  given  price  is  pence,  or  pence  and  farthpjjs, 

.  eyenpjMt  of  a  shilling — Find  <he  value,  of  the 

••'  quantity  as  Is.  per  yard,  &c.  which  divide  by  the 

t  oca  stalling  contained  in  tl-.R  given 

:  fi'kv1  j>;u  Ls  of  tiie  quotif.iT  i\\r  the  remainder  of 

and  the&Min  «i'  ti. ••  cjt.'ifiout.s  will  be 

iswer  in  shillings,  &c.  wl  ce  to  po: 

KXAMF; 

Wl-.at  will  245  Ib.  of  raisins  come  to,  nt  9?d.  pp.r  Ib.  F 

5.      rf. 
4     245     0  value  of  245  Ib.  at  15.  per  pou' 

Jaeof  do.  at  6d.  pci  Ib. 
4       61     S   value  of  do.  at  5d.  JUT  Ib. 
15    3|  value  of  do 

£,0)19,9    OJ 
^n«.  £9  19  Oj  value  of  the  whole  at  S». 


PRACTICE.  US 


372  at  1J  Ms.    2  14    3 
3£5  at  2i  .30  Hi 

827  at  4i  15  10     1} 


576  at    7*  Ms.  18    0  0 
541  at    9k  20  17  0* 

672  at  112  32  18  0 


CASE  V. 

When  the  price  is  shillings,  pence  and  farthings,  and 
not  the  aliquot  part  of  a  pound — Multiply  the  given  quan- 
tity by  the  shillings,  anu  take  parts  for  the  pence  and  far- 
things, as  in  the  foregoing  cases,  and  add  them  together; 
the  sum  will  be  the  answer  ia  shillings. 

EXAMPLES. 

1.  What  will  246yds.  of  velvet  come  to,  at  7s.  3d.  per 
yard  ?  s.  d. 

3d.  |  i  (  246  0  value  of  246  yards  at  Is.  per  yd. 

•  v 


1722  0  value  of  do.  at  7s.  per  yard. 
61  6  value  of  do.  at  3d.  per  yard. 


2,0)178,  3  6 
«S/;s.  £89  3  6  value  of  do.  at  7s.  per  yard. 

ANSWERS. 

s.  rf.                   £.  s.  rf. ; 

Z   What  cost  159  yds.  at    9  10  per  yd.  ?      68  6  10 

3.  What  cfst  146  yds.  at  14     9  per  yd.  ?     107  13     6  ' 

4.  What  cost  120  cwt.  at  11  -  3  per  cwt.  ?     67  10 

5.  What  cost  127  yds.  at    9    8$  per  yd.  ?     61  12  11$ 

6.  What  cost  49|  lb«    at    S  11J  per  Ib.  ?       9  15  Hi 

CASE  VI. 

When  th"  price  and  quantity  given  are  of  several  de- 
n-ominations-^Multiply  the  price  by  the  integers  in  the 
i^  ven  quantity,  and  take  parts  for  the  rest  from  the  price 
of  an  integer ;  which  added  together  will  be  the 
This  is  applicable  to  Federal  Money, 

10* 


114 


TARE  AND  TRETT. 


EXAMPLES. 


1.  "W 

L41b.  of 
Id.  per  c 

2qrs 

iqr. 

14  Ib. 

Ai 
IC.gr 
17      S 
5      1 
14      S 
12      0 
0      0 

tatc 
rais 
wt. 

i 

4 
4 

•*•  : 

S.    4 

1 

2 

ost  5cwt.  3qrs. 
ins,  at  2£.  11s. 
? 
£.      s.      A 
2        11       8 
5 

2.  W! 
8lb.  of  s 
65  cts.  p 

Iqr. 

7  Ib. 
1  Ib.t 

lat 
uga 
ere 

i 

i 

f 

cost  9c\vt.  Iqr. 
•,  at  8  dollars, 
wt.? 
gets. 
8,65 
9 

77,85 
2,1625 
,5406 

,772 

12      18      4 
1       5     10 
12     11 
6      54 

£15       S      6*            ,  Jtns.  280,6303 

'O.                                                                       ANSWERS. 

6  at  g9,  58cts.  per  cwt.      g75,  Glcfs.  Swr. 
0  at  2/.  17s.  per  cwt.                £14  19s.  Sd. 
7  at  01.  13s.  8d.  per  cwt         £10  Qs-  5&- 
7  at  g6,  34cts.  per  cwt.        g76,  47cts.  6m, 
4  at  gll,  91ct8.  per  c\vt.    g2,  55cte.  a^iu. 

TARE   AND   TRETT. 

.1  ARE  and  Trett  arc  practical  Rules  for  deducting 
certain  allowances  which  are  made  by  merchant?,  in 
buying  and  selling  goods,  &c.  by  weight ;  in  which  are 
noticed  the  following  particulars  : 

1.  Gross  Weight,  which  is  the  whole  weight  of  any 
sort  of  goods,  together  with  the  box,  c-^k,  or  bag,  &«. 
which  contains  them. 

Tare,  which  is  an  allowance  made  to  the  buyer 
tor  the  weight  of  the  box,  cask,  or  bag,  &c.  which  con- 
tains the  goods  bought,  and  is  either  afcr'somuch  per  box 
&C. — or  at  so  much  per  cwt.  or  at  so  much  in  the  whole 
§ross  weight. 

3.  Trett,  which  is  an  allowance  of  4  Ib.  on  every  1 0  tl!> 
for  waste,  dust.  &c. 


TARE   AND    1HETT.  H5" 

•1.  C'lfff,  which  is  an  allowance  made  of  2  Ib.  upon 
every  3  c\vt. 

5.  Sutt'tti  is  what  remains  after  one  or  two  allowance? 
have  been  deducted. 

.     CASE  !. 

When  the  question  is  an  Invoice. — Add  the  gross 
weights  into  one  sum  and  the  tares  into  another  ;  then 
subtract  the  total  tare  from  the  whole  gross,  and  the  re- 
mainder will  be  the  neat  weight. 

EXAMPLES. 

1.  What  is  the  neat  weight  of  4  hogsheads  of  Tobacco 
marked  with  the  gross  weight  as  follows : 
C.    tjr.    Ib.  Ib. 

Tare    100 

—  95 

—  83 

—  Si 

359  total  tare. 


J$o.  1  —  9 

0 

12 

2  —  3 

3 

•  4 

3  —  7 

1 

0 

4  —  6 

3 

25 

Whole  gross  52 

0 

13 

Tare  359  lb.=3 

0 

23 

s.  28      3  IS  seat. 
2.  "What  is  the  neat  weight  of  4  barrels  of  Indigo,  No 
•Mvl  weight  as  follows: 

C.  (jr.  Ib.  Ib. 

No.  t  —  4     1     10  Tare    S6"| 

2  —  3     3     02  —      29  [ 

3  —  40     10  —  cwt.  qv.  Ib. 

4  —  40      0  —      35 j  Jlns.  15011 

CASE  II. 

When  the  tare  is  at  so  much  per  box,  cask,  bag,  &c. — 
Multiply  the  tare  of  1  by  the  number  of  bags,  bales,  &c. 
Hie  product  is  the  whole  tare,  v.hich  subtract   from  fhc 
and  the  remainder  will  be  tke  neat  weight. 

EXAMPLES.        i 

1.  In  4  hhds.  of  sugar,  eacli  weighing  lOcwt.  Iqr.  loU). 
gvuss ;  taie  75lb.  v-cr  njid.  how  much  neat  ? 


116  ,   YARK   AKB   TASTT. 

tii't.    grs.  Ib. 

10      1      15  gross  iveig!;t  of  one  Khd. 
4 


41      2        4  gross  weight  of  the  whole. 
75x4=2      2      20  whole  tare. 


Jns.  38      3       12  neat. 

2.  \Vhat  is  the  neat  weight  of  7  tierces  of  rice,  t 
weighing  4cwt.  Iqr.  9lb.  gross,  tare  per  tterce  S41' 

Ans.  28  C.  Oqr.  Zllb. 

5.  In  9  firkins  of  butter,  each  weighing  2qrs.    J 
gross,  tare  11  Ib.  per  firkin ;  how  much  neat  ? 

Jjn.  -1C'.  %r>. 

4.  In  9A\  bis.  of  figs,  each  Sqrs.  19lb.  gri  .  '!b. 
per  barrel;  how  many  pounds  neat?         Jns.  2241.'. 

5.  In  1 6  bags  of  pepper,  each  85lb.  4oz.  gross,  tare  per 

'.i>.  5oz. ;  how  many  pounds  neat  ?      »•/»<>•.  1311. 

6.  In  75  barrels  of  figs,  each  Crjis.  i!£lb.  gross  tare  in 
the  whole,  597lb. ;  how  much  neat  weight : 

sins.  506'.  Iqr. 

7.  "What  is  the  neat  weight  of  15  hluls.  of  Tol)acro.. 
each  weighing  7cwt.  Iqr.  13lb.  tare  lOOlb.  ppr  lihd.  ? 

jffns.  97 C.  Oqr.   1Mb. 

CASE  III. 

"SVhen  the  tare  is  at  so  much  per  cui. — Divide   the 
gross  weight  l>v  the  aliquot  part  of  a  rv  T.  I'm 
which  subtract  from  the  gross  and  tin-  '-r  will  be 

neat  weight. 

EXAMPLES. 

1.  What    is  the  neat  weight  of  44cwt.    Tqr?. 
grow,  tare  14lb.  per  cwt.  ? 

.    C.    qrs.  Ib. 

I  14lb.  |  J  |  44      3  1G    greso. 

5      2  J 


neaf. 


TAUE   AXD   TIIRTT.  117 

2.  What  is  the  neat  weight  of  9  hhtk.  of  tobacco,  each 
weighing  gross  Scwt.  Sqrs.  14lb.  tire  I61b.  per  cwt.  ? 

.flws.  68czt-£.  l//r.  24i'&. 

3.  What  is  the  neat  weight  of  7  bbls.  of  potash,  each 
\vcighing  29llb.  gross,  tare  10'ib.  per  «wt.  ? 

jJns.  1281ft.  6oz. 

4.  In  25  barrels  of  figs,  each  Scwt.   Iqr.  gross,  tare 
per  cwt.  I61b.  j  how  much  neat  weight? 

Jkns.  48cwt.  24ft. 

5.  In  83cwt.  3qrs.  gross,  tare  20lb.  per  cwt.   what 
neat  weight  ? 

Jlns.  GScwt.  Sqrs.  5lb. 

6.  In  45cwt.  Sqrs.  2llb>  gross,  tare  8lb.  per  c\vt.  how 
much  neat  weight  r 

Ans.  42cw>£.  2qrs.  i7ift. 

7.  What  is  the  value  of  the  neat  weight  of  8  hhds.  of 
sugar,  at  g9,  54cts.  per  cwt.  each  weighing  lOcwt.  Iqr. 
141b.  gross,  tare  14lb.  per  cwt.  ? 

Jlns.  R092,  S4c#s.  2Jtn. 


CASE  IV. 

When  Trett  is  allowed  with  the  Tare. 

1.  Tint]  the  tare,  which  subtract  from  the  grass,  anil 
call  the  remainder  suttle. 

2.  Divide  the  suttlo  by  26.  and  the  quotient  will  be  the 
trett«,  which  subtract  from  the  suttle,  and  the  remainder 
wiH  be  the  neat  weight. 

EXAMPLES. 

1.  In  a  hogshead  of  sugar,  weighing  lOcwt.  Iqr.  12ft). 
gross,  tare  14lb.  per  cv.  t.  tiett  4lb.  per  104lb.*  how 
much  neat  weight  ? 

*  This  is  the  Irett  allowed  in  London.  Tketreason  of 
dividing  by  26  is  because  4ft.  is  ^  of  104Z&.  but  if  the 
trett  <'.-  at  any  other  rate,  other  parts  must  be  taken,  ac- 
cording to  tf.e  rate  proposed,  Sfc. 


J1&  TARF.   AND 


Or  thus 

cwt.  qr.  Ib.  cict.  gr.   fti. 

10     1     12        14ll>=})10     1     1C  c; 
4  115  tare. 


26)9    0      7  sultle. 
111  trett. 


3ns.  8    2    24  neat. 


M=J.)1160  gross, 
145  tare. 


26)1015  suttle. 
39  trett. 

Stts.  97&lb.  neat. 

2.  I?i  9  c\vt.  2  qrs.  IT  Ib.  gross,  tare  41  Ib.  trett  4  Ib. 
per  104  Ib.  how  much  neat?       Jlns.  8c?rf.  S^r.s.  20/6. 

3.  In  1J  chests  of  sugar,  weighing  117c\vt.  21  Ib.  gross, 
tare  1T3  Ib.  trett  4  Ib.  per  104,  how  many  cwt.  neat  ? 

Jlns.  lllctct  22/6. 

4.  "What  is  the  neat  weight  of  3  tierces  of  rice,  each 
weighing  4  cwt.  S  qrs.  1-4  Ib.  gross,  tare  16  Ib,  per  cwt 

ing  trett  as  usual  ? 

Jlns.  IQcict.  Qqrs.  6lb. 

5.  Tri  25  iiarri'ls  <>t  figs,  each  84  Ib.  gross,  tare  12  Ib. 
per  <  ..  t.  trett 4  Ib.  j»cr  104  Ib. ;  how  many  pounds  neat  r 

'3  + 

•  value  of  the  neat  weight  of  4  b.incU 
>';  uumbers,  weights;  anil  allovanrv* 
as  ii  per  pouii'i  f1 

cict.  /JTS,  /A. 
No  1    (i.n^     I     2     l.n 

2  1     0  Tare.  10  Hi.  ] 

S  1     0    09  f     Trett  4  !b.  ,MT  10- 

4  tt    r?     21 


TARE     4ND    TRETT.  119 

CASE  V. 

When  Tare,  Trett,  and  Cloff'  are  allowed  : 
Deduct  the  tare  and  trett  as  before,  and  divide  tli-e  sut- 
tle  by  168  (because  2  Ib.  is  the  T]-g   of  3  cut.)  the  quo- 
tient will  be  the  cloflf,  which  subtract  from  the  suttle,  and 
the  remainder  will  be  the  neat  weight. 


EXAMPLES. 

1.  In  3  hogsheads  of  Tobacco,  each  weighing  13  cwt. 
3  <irs.  23  Ib.  «;ross,  tare  107  Ib.  per  hogshead;  trett  4  Ib. 
per  104  Ib.  and  cloflf  2  Ib.  per  3  cwt.  as  usual ;  how  muth 
neat. 

cict.  qrs.  Ib. 
13     3     23 
4 


1563  Ib.  gross  of  1  hlid. 
3 

4689  whole  gross. 
107X3=  321  tare. 

26)4368  suttle. 
168  trett. 

168)4200  suttle. 
25  cloft*. 


•ffns.  4175  neat  wei;'3'hi. 

2,  THiat  is  the  neat  v.viuht  of  26  cut.  3  qrs.  CO  Ib 
tare  02  Ib.  the  allowance  of  trett  and  clofl'as  usur.l  ? 

Jins.  neat25cu.'t.  lyr.  5lb.  loz.  nearly;  omit t if-: 

further  fraction*. 


HO  INTEREST. 

INTEREST. 

INTEREST  is  of  two  kinds ;  Simple  and  Compound. 
SIMPLE  INTEREST. 

Simple  Interest  is  the  sum  paid  by  the  borrower  to  the 
lender  for  the  use  of  money  lent ;  and  is  generally  at  a 
certain,rate  per  cent,  per  annum,  which  in  several  of  the 
United  States  is  fixed  by  law  at  6  per  cent,  per  annum  ; 
that  is,  6/.  for  the  use  of  100/.  or  6  dollars  for  the  use  of 
100  dollars  for  one  year,  &c. 

Principal,  is  the  sum  lent. 

Rate,  is  the-  sum  per  cent,  agreed  on. 

Amount,  is  the  principal  and  interest  added  togetheis 

CASE  I. 

To  find  the  interest  of  any  given  sum  for  one  yva4 
RULE. 

Multiply  the  principal  by  the  rate  per  cent,  and  divide 
*he  product  by  100;  the  quotient  will  be  the  answer* 

EXAMPLES. 

1.  What  is\he  interest  of  S9J.  lls.  8}d.  for  one  year, 
at  G:.  per  cent,  per  anmm  ? 

£.     5.     d. 
S'J     11     8* 
6 


10   s 

20 


IS 

6]  03 
4 

OJ12  Jtns.  £4  7s.  6J.^A 

•2.  What  is  t'ie  interest  of  £S6f.  10s.  4d.  for  a  jf«-ar,  at 
5  jer  cent  ?  AM.  &  I L  I6t.  CJ. 


SIMPLE  INTEREST.  i&l 

&    V.  i>at  is  the  interest  «f  571?.  1 3s.  9d.  for  one  year, 
at  62.  per  cent.  ?  .fln?.  £34  6s.  Oirf. 

4.  What  is  the  interest  of  2/.  12s.  9$d.  for  a  year, at 
6^.  per  cent.  ?  tins.  £0  3s.  2rf. 

FEDERAL  MONEY. 

5.  \Vhat  is  the  interest  of  468  dols.  45  cts.  for  one  ye&r 
at  6  per  ce^t.  ?  $   cts. 

468,  45 
6 


281,10,  70=g28,  lOcfs.  7m.  Ansf 

Here!  cut  off  the  two  right  hand  integers,  which  di- 
vide by  100  :  but  to  divide  federal  money  by  100,  you 
need  only  call  the  dollars  so  many  cents,  and  the  inferior 
denominations  decimals  of  a  c«nt,  and  it  is  do:ie. 

Therefore  you  may  multiply  the  principal  by  the  rate, 
and  place  the  separatrix  in  the  prociuct,  as  in  multiplica^ 
tion  of  federal  money,  and  all  the  figures  at  the  left  of 
the  separatrix,  will  be  the  interest  in  cents,  and  the  first 
figure  on  the  right  will  be  miffs,  and  the  others  decimab 
of  a  mill,  as  in  tbe  following 

EXAMPLES. 

6.  Required  the  interest  of  155  dels.  25  cts.  forayeai 
at  6  per  cent.  S  cts. 

155,  25 
6 


811,  50=»8,  llcfr.  5."i.  Jlns. 
7.  "\Vhatis  tin-  interest  of  19  dulars  51  cents  t 
year  at  5  per  cent.  ?  $   cts. 

19,  51 
5 


97,  55=f>7rf3.  5$ir 

8.  What  is  the  interest  of  436  doi.at?  for  or.o  v. 
6pei*rcnt.  f  6 

Jlns. 


1-22  SIMPLE   INTEREST 

ANOTHER  METHOD. 

Write  down  the  «nvep  principal  in  cents,  which  multi- 
ply b)  the  rate,  and  divide  by  100  as  before,  and  you  will 
have  the  interest  for  a  year,  in  cents,  and  decimals  of  a 
cent,  as  follows : 

9.  What  is  the  interest  of  ,g73,  65  cents  for  a  year,  at 

6  per  cent.  ? 

Principal  7365  cents. 
6 

*  *  * 

.5ns.  441,90cte.:=44lTyte.  or.g4,41cfs.  9m. 

10.  Required  the  interest  of  £80,  45cts.  for  a  year,  at 

7  per  cent.  ?  Cents. 

Principal  8545 

7 


Arts.  598,  15  crnfs,=S5,98cfs.  Ijm. 
CASE  II. 

To  find  tlie  simple  interest  of  any  sum  of  money,  for  any 
number  of  years,  and  parts  of  a  year. 

GENERAL  -RULE. 

1st.  Find  the  interest  of  the  given  sum  for  one  year, 
iid.  Multiply  the   interest  of  one  year  by  the*  given 
number  of  years,  and  tiie  product  will  be  the  answer  for 
that  time. 

Gd.  If  there  be  parts  of  a  year,  as  months  and  days, 

'lie  moiii'  (tnd 

of  Three  iiiroct.  •  ing 

SO  days  to  the  mouth,  ;  ./.juut  part.:   uf  the 

-same.* 


*  By  a!!mvii  .i*juot 

,  r.rJ.iua.y  sum 
:u  be  very 

.f  tluvi  :  '•  t\ 


.     SIMPLE   INTEREST. 


123 


EXAMPLES. 


1.  What  is  the  interest  of  751.  8s.  4d.  for  5  years  and 
2  months,  at  61.  per  cent,  per  annum  ? 
£.     s.    d. 


75     8      4 


4)5.2    10 
20 


£.   s.  d. 
2mo.=|)4  10  G  Interest  for  1  year. 


22  12  6  do.  for  5  years. 
0  15  1  do.  for  2  months. 

£23     77  Ms. 


2.  What  is  the  interest  of  64  dollars,  58  cents,  for  3 
years,  5  months,  and  10  days,  at  5  per  cent.  ? 
864,58 
5 


322,90   Interest  for  1  year  in  cents,  pw 
3  [Case  I. 


4  mo.  ^ 
1  mo.  J 
lOdays,^ 


968,70  do.  for  3  years. 
107,63  do.  for  4  months. 
26,90  do.  for  1  month. 
8,96  do.  for  10  days. 


Ms.  1112,19=1112c£s.  or  gll,  12c.  1TV». 
S.  What  is  the  interest  of  789  dollars  for  2  years,  at  6 
per  cent.  ?  Ms.  §94,  tiScfs. 

4.  Of  37  dollars  50  cents  for  4  years  at  6  per  cent,  per 
annum  ?  Ms.  flGUcfcj.  or  g9 

5.  Of  325  dollars  41  cents,  for  3  years  and  4  months, 
at  5  per  cent.  ?  Ms.  £54,  23cfe.  5;n. 

6.  Of  325Z.  12s.  3d.  for  5  years,  at  6  per  cent,  f 

Jus.  £97  13s.  Sd. 

7. 'Of  174?.  10s.  6d.  for  3  and  a  half  years  a; 
cent.  ?  Jliis.  £ 

8.  Of  150?.  16s.  8d.  far  4  years  and  7  months,  ai 
cent?  Ms.  £41  9s.  ~d. 


SlMl'LE    INTEREST. 

!\  Of  1  dollar  ft'i-  1£  years  at  5  per  eent.  ? 

..3ns.  6Qcts. 

10.  Of  -2]  5  dollars  34  cts.  for  4  and  a  half  years,  at  3 
ami  a  half  per  cent.  ?  .ins.  g33,  9 Ids.  6m. 

11.  What  is  the  amount  of  324  dollars,  61  cents,  far 5 
%  ears  and  5  months,  at  6  per  cent.  ? 

•0ns.  8430,  Wets,  bf-fom. 

12.  "What  will  SOOOJ.  amount  to  in  12  years  and  10 
months,  at  6  percent.?  .i/zs.  £5310. 

13.  What  is  the  interest  of  25 7/.  5s.  Id.  for  1  year  and 
:>  quarters,  at  4  per  cent.  ?        jlns.  £18  05.  id.  Sqrs. 

14.  W  hat  is  Hie  interest  of  27'.)  dollars,  87  oents  for  2 
years  and  a  half,  at  7  per  cent,  per  annum  ? 

»fns.  £48,  9rr/a.  7im 

15.  What  will  279Z.  13s.  8d.  amount  to  in  3  years  and 
.-.  half  at  5}  per  cent,  per  annum  ? 

Ans.  £331  15.  6rf. 

If-.  What  Is  the  amount  of  341  dols.  60  cts.  for  5 yeas* 
a:ul  3  quarters,  at  7  and  a  half  per  cent,  per  tinnum  ? 

.flKs.  S488,  9lic*s. 

17.  What  will  730  dols.  amount  to  at  6  per  cent,  iu  .5 
years,  7  months  and  1C  Jays,  or  ^2T  of  a  year  ? 

.dtos.  8975,  99cte. 

18.  What  is  the  interest  of  1825/.  at  5  per  cent,  per 
annum,  from  March  -Uh,  179(5,  to  March  29th,  1799,  (olr 
lowing  thc^-ear  to  contain  363  days  ?)         .']  zs.  £280. 

No  --E. — The  Rules  for  Simple  Interest  serve  also  to 
calculate  Commission,  Brokerage,  Insurance,  or  any  thing 
else  estimated  at  a  rate  per  cent. 


COMMISSION, 

IS  an  allowance  of  so  much  per  cent.  t«  a  factor  or  cor- 
respoadoni  abroad,  for  buying  and  selRng  goods  for  hi« 
•ycr. 

KXAMPLES. 

1.  What  will  the  commission  of  843/.  10s.  come  to  at 
5  per  cent.  ? 


$iMPL.E   INTEREST.  125 

Or  thas, 
£.    s. 
£  5  is  ^43  10 

421  17  10  .fas.  £42  3  6 

20 

5|50 
12 


6|00  £42  3s.  6d. 

2.  Required  the  commission  on  964  dote.  90  cte.  at  2$ 
percent?  .8ns.  821,  71cte. 

5.  What  may  a  factor  demand  on  1 1  per  cent,  commis- 
sion, for  laying" out  3568  dollars  ?  Jlns.  g62,  44cfs. 


BROKERAGE,  - 

IS  an  allowance  of  so  much  per  cent,  to  persons  assist- 
ing merchants,  or  factors,  in  purchasing  or  sell  ing  goods. 

EXAMPLE'S. 

1.  What  is  the  brokerage  of  750?.  8s.  4d.  at  6s.  8d. 
per  cent  ? 

£.    s.    d. 

750    8    4        Here  I  first  find  the  brokerage  at  1 
1  pourvi  per  cent,  and  then  For  the 

given  rate,  which  is  -£  t>f  r  pound. 

7,50     8    4 

20  S.  d.         £.  s.  d.  qrs. 

6  8=£)7  10  1 

10,08 

12  Ans.  £2  10  0  Ij 

1,00 

2.  \Viiatistrtebrokerageupon  4125  dols.  at  £  or  75 
eents  per  cent.  ?  Jlns.  S530,  93ct$.  7im. 

S.  it"  a  broker  sells  goods  to  the  amount  of  5COO  dels, 
what  is  his  demand  at  65  cts.  per  cent,  r 

Ans.  g3i2,  50cf« 
1)  * 


'i2G  SIMPLE    INTEREST^ 

4.  What  may  a  broker  demand,  when  he  sells  goods  to 
the  value  of  508f.  ITa.  lOd.  and  I  allow  him  1$  per  cent  ? 

£7  13*  8& 


IS  a  premium  at  so  much  per  cent,  allowed  to  persons 
and  offices,  for  making  good  the  loss  of  ships,  houses,  mer- 
chandize, &c.  which  may  happen  from  storms,  fire,  &c. 

EXAMPLES. 

1.  What  is  the  insurance  of  7251.  8s.  lOd.  at  12$  per 
cent,  r  Ans.  £90  13s.  7±d. 

2.  \Vhai  is  the  insurance  of  an  East-India  ship  and 
cargo,  valued  at  125425  dollars,  at  15$  per  cent.  ? 

Ans.  g!9130,  ZTcts.  5m. 

3.  A  man's  house  estimated  at  5500  dollars,  was  insu- 
red against  fire,  for  1$  per  cent,  a  year :  what  insurance 
djil  lie  annually  pay?  Jbis.  &61,  25cfs. 


SHORT  PRACTICAL  RULES, 

Anting  Interest  at  G  per  cent,  either  f 
or  months  and  days. 

I.  FOR  STERLING  MONEY. 
RULE. 

1.  If  the  principal  consists  of  pounds  only,  cut  off  tk« 
unit  figure,  and  as  it  then  stands  it  will  be  the  interest  for 


,  &c.  re- 
decimal 

point  one  place,  or  figure,  further  towards  the  left  hand, 
and  as  the  decimal  then  stands,  it  will  shew  the  interest 
e  month,  in  .-hillings,  and  decftnals  of  a  shilling. 

EXAMPLES. 

i.  Required  the  interest  of  541,  for  seven  months  ana 
'.y«,  ;ir  6  per  cent 


-SIMPLE,    INTEREST,  127 

S. 

10  dajs=fj5,4  Interest  for  one  month. 
7 

57,8  ditto  for  7  months, 
1,8  ditto  for  10  days. 

'  [Ans.  39,6  shillings  =£1  19s.  7,2rf. 
12 

7,2 

fi.  What  is  the  interest  of  42/.  10s.  for  11  months,  at  6 
per  cent.  ? 

£•    s-       £. 

42  10  =  42,5  decimal  value.   m 
Therefore  4,25  shillings  interest  for  1  mouth. 
11 

£.  s.  J. 

Ms.  46,75  Intercstforllmo.  =  269 


3.  Required  the  interest  of  942.  7s.  6d.  TITOL 
five  months  and  a  half,  at  6  pe  :am. 

4.  What  is  the  interest  of  19.1.  lt>s.  for  one  third  of  a 
month,  at  6  per  cent.  ?  -.  5,}6d. 

II.  FOR  FEDERAL  MONEY. 
RULE. 

1.  Divide  the  principal  by  2,  placing  the  sepaiatrr<;  a* 
usual,  and  the  quotient  will  be  flic  interest  for  oneirr 
in  cents,  anu  decimals  of  a  cent;  that  is,  die  I'-. 
the  left  of  the  separatrix  will  be  cents,  and  those  u:. 
right,  decimals  of  a  cent. 

£.  Multiply  the  interest  of  one  month  by  thegiven  r\\>r.\. 
ber  of  months,  or  months,  and  decimal  pits  thereof,  or  fur 
tl^  tirjys  tike  the  eveu  parts  of  a  month,  &c. 


1-28  SIMPLE  INTEREST: 

EXAMPLES. 

1.  What  is  the  interest  of  341  dols.  52  cts.  for  74  months  ? 
2)341,52 

Or  thus,  170,76  Int  for  1  month. 
170,76  Int.  for  1  mouth.         x7,5  months. 


853SO 


1 195.32  do.  for  7  mo.         1 19552 
85,58  ilo.  for  •*  mo.  &  cts.m. 

1280,700ds.  =12,80  7 

1*280,70  ,9Ms.  1280,7cL>;.=S12,  SOcfs.  7m. 

2.  Required  tlve  interest  of  10  dols.  44  cts.  for .3  years, 
5  months  anil  10  days. 

2)10.44 

10  days  —  :)     5,22  Interest  for  1  month. 
41  months. 

5,22 
208,8 


214,02  ditto  for  41  months. 
1,74  ditto  for  10  days. 

2f5,76cf.«.      .£«<;. =g2,  15cfs.  7m. -f 
<  i.ai.  is  the  interest  of  342  dollars  for  11  months  ? 

The  J  is  171  Interest  for  one  month. 
11 

JRS.  1881cfs.=gl8,  81c<*. 

No  IK. — To  find  the  interest  of  any  sum  for  2  months, 
•ic.r  cent,  you  need  only  call  the  dollars  so  many  cents, 
:thd  Lhe  inferior  denominationsi  decimals  of  a  cent,  and  it 
i»  douo  :  Thus,  the  interest  of  100  dollars  for  two  months, 
is  100  ronts,  or  1  dollar:  anrl  S^J,  40  cts.  j*  25  cts.  4  in. 
&c.  v.  Ill-  li  tnvL's  the  following 

RULE  II. 

Miiltijily  the  principal  by  half  the  number  pf  months, 
i  in-  product  will  shew  the  interest  for  the  given  time. 
ufs  and  decimals  of  a  cent,  as  above. 


SIMPLE    INTEREST.  129 

EXAMPLES. 

1  Required  the  interest  of  3 16  dollars  fer  Ij'ear  and 
10  months.  1 1  =half  the  number  of  mo. 


Ans.  3476c£s.=gS4,  76c*s. 
2.  What  is  the  interest  of  364  dols.  25cts.  for  4  months  ? 

8    cts. 
.  S64,  25 

2  half  the  months. 


728,  50c*s.  ^ns.=gf,  28c£s.  5m. 

III.  When  the  principal  is  given  in  federal  money,  at 
6;per  cent,  to  find  how  much  the  monthly  interest  v,  ill  be 
in  New-England,  &c.  currency. 

RULE. 

Multiply  the  given  principal  by  ,03  and  the  product 
will  be  the  interest  tor  'one  month,  in  shillings  and  deci- 
mal parts  of  a  shilling. 

EXAMPLES. 

t.  What  is  the  iaterest  of  325  dols.  for  11  months  ? 
,03 


9,75  shil.  int.  for  1  ITK 
Xll  months. 

.  107,25s.  =£5  7s.  Srf. 
&  What  is  the  interest  in  New-England  currency,  of 
31  dols.  68  cts.  for  5  months  ? 

Principal  31,68  dols. 
,03 

,050-i  Interest  for  one  month. 
5 

Ms.  4,7520s. =4s.  9d. 
12 

90840 


SIMl'LE    i:,'TE,RKST. 


IV.   \  principal  is  given  in  pounds,  shillings, 

tr-Knglanu   <  im-ency,  ;il  G  per  cent,  to  find   how 
.aoiithiy  interest  will  be  hi  federal  money. 

RULE. 


lv  (he  pounds.  &r.  by  5,  and  divide  that  pro- 
due1  <[unfifMt  will  be  the  interest  for  one  month, 

•  It'ciiuals  of  a  cent,  &c. 


EXAMPLES. 

i.  A  note  Ink-  £411  New-England  currency  has  beea. 
on  interest  one  month  ;  how  much  is  the  interest  thereof 
in  federal  nionr-.  r  ,'\ 

411 


)2055 


.2.  Required  the  interest  of  39/.  18s.  N.  E.  curretfcj, 
for  7  montln  ?  £. 

59,9  decimal  value. 
5 


Interest  for  1  mo.      60,5  cents. 

7 


Ditto  for  7  months,  465,5cfs.=g4,  65c/s.  5m.  «3ns.. 


V.  When  tlw  principal  i-  •>  Nr.w-En^hind   and 

Virginia  cmrency,  at  6  per  cent,  to  iind  Ihe  rVjterest  for 
,  in  dyllur.s,  cents  and  mills,  by  inspection. 

RULE. 

Since  Ike  interest  of  a  year  will  be  ju^t  so  many  cents 

•  ^ivcTi  prinripal  contains  shillings,  therefore,  \vrite 

<lo\vn  ihe  ghiiliug^  and  call  (licin  i  <he  pence  in 

ilie  priiuipul   luadc  less  bv  1   if  they  exceed  f-,  or  by  2 

•  frd  9,  \vi!l  be  the  mills,  ver\  rearlv. 


SIMPLE   INTEREST. 

EXAMPLES. 

1.  What  is  the  interest  of  2/.  5s.  for  a  year  at  6per  ct.? 

£2  5s.=45s.  Interest  45cts.  the  dusirer. 

2.  Required  the  interest  of  100/.  for  a  year  at  6  per  ct.? 

£100=2000s.  Interest  200()cte.=g20  Ans. 

3.  Of  27s.  6d.  for  a  year  ? 

Jins.  27s.  is  27cfs.  and  Gd.  is  5  mills. 

4.  Required  the  interest  of  5l.  10s.  1  Id.  for  a  year  ? 

£5  10s.=110.s./»femn;0cfc.=81,  lOcts.  Om. 
11  pence — 2  per  rule  leaves  9=  9 

Jus.  SI,  10       9 

VI.  To  compute  the  interest  on  any  note  or  obligation, 
when  there  are  payments  in  part,  or  indorsements. 
RULE. 

1.  Find  the  amount  of  the  whole  principal  for  the  whole 
time. 

2.  Cast  the  interest  on  the  several  payments,  from  the 
time  they  were  paid,  to  the  time  of  settlement,  and  find 
iheir  amount ;  and  lastly  deduct  the  amount  of  the  seve- 
ral payments,  from  the  amount  of  the  prim ' 

EXAMPLES. 

Suppose  a  bond  or  note  dated  April  17, 179.1,  was  given 
for  675  dollars,  interest  at  6  per  ceait.  and  there  \ver« 
payments  indorsed  upon  it.  as  follows. 
"First  payment,  148  dollars,  May  7,  1794. 
Second  payment,  341  dols.  August  17,  17$r>. 
Third  payment,  99  dols.  Jan.  2,  1793.     I  domaini  lio\v 
much  remains  due  un  said  note,  the  17th  of  June,  1798  : 
8   cts. 

14S,  00  fi!-;t  payment.  May  7,  1.  JV. 

36,  50  interest  up  to— June   17,  ITt'S. -i       14 

184,  50  amount 


341,  00  second  pay  me     ,  AM-.:.  17. 
57,  51  Interest  to June  If, 


S*8,  51  amount. 


SIMPLE  INTEREST. 


s. 

99,  00  third  payment,  January  2,  1798. 
2,  72  Interest  to—  June  17,  1798.=5imo. 

101,  72  amount. 

184,  50"| 

378,  5  1  v  several  amounts. 

101,  72  J 

664,  73  total  amount  of  payments. 

075,  00  note,  dated  April  17,  1795.       Fr.  ?n«. 
209,  25  Interest  to—  June  17,  1798.  =5      2 


884,  25  amount  of  the  note. 
664.  73  amount  of  payments. 

J&219,  52  remains  due  on  the  note,  June  17,  1798. 
2.  On  the  iCth  of  January,  1795, 1  lent  James  Paywell 
500  dollars,  on  interest  at  6  per  cent,  which  I  received 
back  in  the  fol lowing  partial  payments,  as  under,  viz. 
1st  of  An; -U,  l/9'.i        -  g  50 

16th  of  July,  1797  -  -    400 

1st  of  Sept.     1798        -  -  60 

How  stands  the  balance  between  us,  on  the  16th  No- 
VCiriber,  1800?  .  due  to  me  g63,  18cfs. 

3.    A  PKOMISSORY    NOTE,   VIZ. 

£62  10.;.  .Yrir- London,  April  4,  1797. 

On  demand  I  promise  to  pay  Timothy  Careful,  sixty- 
two  pounds,  ten  shillings,  and  interest  at  6  per  cent,  per 
annum,  till  paid;  value  received. 
JOHN  STANUY,  PKTKtt  PAYWELL. 

RICHARD  TESTIS. 

Indorsement*.  £.   s. 

1st.  Received  in  part  of  the  above  note,  Sep- 
tember 4,  1799.  50    0 
And  payment  .^ une  -1.  1800,  12  10 
How  much  remains  due  or,  said  note,  the  fourth  day  of 
December,  J800  ?                                             £.    s."  «;. 

^n»,  9    19.    <> 


SIMPLE   INTEREST.  •       133 


NOTE.  Tha  preceding  Hula,  by  custom  is  rendered  so 
popular,  and  so  much  practised  and  esteemed  by  many  on 
account  of  its  being  simple  and  concise,  that  I  have  given 
it  a  place :  it  maij  answer  for  short  periods  of  time,  but  in 
a  long  course  of  years  it  will  be  found  to  be  very  erro- 
neous. 

Although  this  method  seems  at  first  view  to  be  upon  the 
ground  of  simple  interest,  yet  upon  a.  little  attention  the 
following  objection  will  be  found  most  clearly  to  lie  against 
it,  viz.  that  the  interest  will,  in  a  course  of  years,  com- 
pletely expunge,  or  as  it  may  be  said^at  up  the  debt.  For 
an  explanation  of  this,  take  the  following 


EXAMPLE. 


A  lends  B  100  dollars,  at  6  per  cent,  interest,  ami 
takes  his  note  of  hand ;  B  does  no  more  than  pay  A  at 
every  year's  end  6  dollars,  (which  is  then  justly  due  to 
B  i'or  the  use  of  his  money)  and  has  it  endorsed  on  his 
note.  At  the  end  of  10  years  B  takes  up  his  note,  am? 
the  sum  he  lias  to  pay  is  reckoned  thus  :  Th<4  principal 
100 dollars, on  interest  10  years  amounts  to  160  dollars; 
there  ate  nine  endorsements  of  6  dollars  each,  upon 
which  the  debtor  claims  interest ;  one  for  9  years,  the 
sec  uid  for  8  years,  the  third  for  7  years,  and  so  down  to 
the  time  of  settlement;  the  whole  amount  of  the  several 
endorsements  and  their  interest,  (as  any  one  can-see  by 
casting  it)  is  S^O,  20cts.  this  subtracted  from  160  •'• 
the  amount  of  the  debt  leaves  in  favour  of  the  creditor, 
889,  40cts.  or  glO,  20  cts.  less  than  the  original  princi- 
pal, of  which  he  has  net  received  a  cent,  but  only  its  an- 
nual interest. 

If  the  same  note  should  lie  £0  years  in  the  same  way, 
B  v/ould  owe  but  57  dols.  60  cts.  without  paying  tlie 
least  fraction  of  the  100  dollars  borrowed. 

Extend  it  to  £8  years,  and  A  the  creditor  would  full 
in  debt  to  B  without  receiving  a  cent  of  the  100  do! 
•which  lie  lent  him.     Seo  a  better  Ilule  in  Simple  Inter - 
wt  by  Decimals,  page  175. 


C 0 .NJ I1 0 L  N  »  I XT £ H E S T. 

COMPOUND  INTEREST, 

Ih  \vlicu  the  interest  :s  added  to  the  principal,  at  the  end 
of  the  year,  and  on  that  amount  the  interest  cast  for  anoth- 
er year,  and  added  again,  and  soon  :  this  is  called  Inter- 
rst  upon  Interest.  % 

RULE. 

ind  ir.c  interest  for  a  year,  and  add  it  tor  the  principal, 
.  hich.ca!l  the  amount  Tor  the  first  year  ;  find  the  interest 
,is  amount,  which  add  as  before,  for  the  amount  of'the 
:ul,  and  so  on   for  auv  numb?.r  of  years  required. 
-Subtract  the  original  principal  from  the  lust  amount.  and 
the  remainder  will   be  the  Compound  Interest  fu. 
\vliole  time. 


EXAMI'LES. 


1.  Required  the  amount  of  100  dollars  for  3  years  at  6 
per  cent,  per  annum,  compound  interest  ? 

8  c/s.  g  cts. 

1st  Principal  100,00  Amount  1(»()5')0  for  1  year. 
c.d    Principal  106,00  Amount  i  1:2,36  for  2  years. 
3d  "Principal  112,36  Amount  119,ioi6ior  3yns.  Jlns. 

2.  What  is  the  amount  of  425  dollars,  for  4  years,  at  5 
per  cent,  per  annum,  compound  interest  ? 


5.  What  will  400^.  -,  in  4  yeais,  at  6  per  cent. 

liar  annum,  compound  iin.e*i  ;•'.<  r  .--Ins.  /,'.>H4  \i)s.  9$d. 

4.  What  is  the  coir-  pound  interest  of  liiO/.  10s.  for  S 
yen,  ct.  per  lusnum  ?      •i/^.  £28  14s.  11]^.+ 

5.  \Viiat  is  the  cnmpouftd  interest  ot  500  dollars  for  4 
years,  at  <•  per  cent,  per  anr.um  r  Jlns.  gli>l,258-f- 

h;tt  will  '1000  dollars  amount  to  in  4  years,  at  7 
per  cent  pi-r  annum,  c-> 

;310,  T9ds.  fiia.-f 

7.  What  is  the  amount  of  7.:  )  •,-  4  j>  eats,  at  6 

,j    r  annum,  compound  inter. 
•Jin: 

8.  Wlint  is  the  compound  isii.fi  est  of  870  «, 
for  Si  years,  at  G  per  vei.t.  nei  annum  r 


01! 

DISCOUNT,  % 

l^>  an  allowance  Tingle  for  the  payment  of  any  sum  of 
money  before  it  becomes  due;  or  upon  advancing  ready 
Money  for  notes,  bills,  ,£<c.  which  are  payable  at  a  future 
.  "  What  remains  al'ter  the  discount  is  deducted,  is  the 
present  worth,  or  such  a  sum  as,  if  put  to  interest,  would 
at  the  given  rate  and  ti  given  su. 

debt. 

RULE.  • 

As  the  amount  of  IGOZ.  or  100  dollars,  at  the  given  rat-c- 
lime :  is  to  t!;e  interest  of  100,  at  the  same  rate  and 
time  :  :  so  is  t!<; 

Subtract  t  .  en  sum,  and  the  re- 

mainder i.~ 

Or—,  •  of  100  :  is  to  100  ::  so  is  the 

H  sum  or  <'-.-;>t  :  to  the  present  wort!). 

PROOF. — Find  the  amount  of  -i>t  worth,  at  the 

given  rate  and  tune,  and  if  the  work  is  right,  that  will  be 
equal  to  the  given  sum. 


1 .  What  must  be  discounted  for  tlie-ready  payment 
)()  dollars,  due  a  year  hence  at  (J  per  cent,  a  year  ? 


S        S  S       S  e.ts. 

As  106  :  (i  :  :  100  :  5  66  the 
100,00  given  sum. 
5,06  discount. 


•  v-t  the  present  worth. 

2.  "What  sum  in  ready  money  will  discharge  a  debt  <>: 
925?.  due  1   vearand  b  montlis  hct'ie.  at  G  [-er  c 
£100 

10  Interest  for  £0  mo:: 


110  Ain't.  /;.        £.  £.        £.    s.  d. 

As  110  :  100  :  :  \l-25  :  840  18  2+Jns. 
f>.  AVIiat  is  tlie  present  worth  of  600  dollars,  due  4 
years  hence,  at  5  per  ceat.  ?  «2ns.  g500 

4.  What  is  the  discount  of  275/.    10s.  for  10  m< 
at  ^5  m-r  tent,  per  annum  r  •  £  I  3  2s 


ANSUIilES. 

5.  BougLi  goods  amounting  to  G15  dols.  75  cents,  at  7 
months  creuit;  how  much  ready  money  must  I  pay,  <iis- 
count  ;it4i  per  cent,  per  annum  ?  Jlns.  go'OO. 

6.  'What  sum  of  ready  money  must  be  received  for  a 
bill  of  900  dollars,  due  73  days  hence,  discount  at  (>  per 
cent,  pfer  annum  ?  "  »3/i5.  &S89,  32cte.  8'n. 

NOTE. — When  sundry  sums  are  to  be  paid  at  different 
times,  find  {he  Rebate  or  present  worth  ot  each  particular 
payment  separately,  and  when  so  found,  add  them  into 
one  sum. 

EXAMPLES 

7.  What  is  the  discount  of  7.'>0/.  the  one  half  payable 
in  six  months,  and  the  other  half  in  six  months  after  that, 
at  7  per  cent.  ?  .flits.  £37    IQs.  Z$d. 

8.  If  a  legacy  is  left  me  of  2000 'dollars,  of  which  500 
dols.  are  payable  in  6  months.  800  dols.  payable  in  1  year, 
ami  the  rest  at  the  end  of  5  years ;  how  much  ready  money 
ought  \  to  receive  for  said  legacy,  allowing  6  per  cent, 
discount?  Ans.  81833,  37c?s.  4m. 


ANNUITIES. 

AN  Annuity  is  a  sum  of  money,  payable  every  year,  or 
for  a  certain  number  of  years,  or  forever. 

\VI:cn  the  debtor  l<e:-y>i  the  annuity  in  his  own  hands, 
beyond  t  !  to  be  in  arrears. 

Th"  '••rliic  li;ue  thi-y  h;n 

.teix>st  due  on  each,  is  called 
th:'  ; 

(fan  o!l',  or  pai;l  all  at  o 

.•  of  tin-  !;  -  :he  puce  v.hich  is  ^lid  i 

ii  c,:  led  liic  prtv.ent  \vorth. 

To  find  t'ue  amount  of  an  annuity  at  - 

RU 

I.  Fin-.l  ilie' interest  oi'  the  givrn  an'Hiity  1 
n..    \nd  tlien  for  2,  3,  &.c.  yi'.:  ,  time, 

'".'tip!  y  the  annuity  by  the  n'n.ibi 
i  l;l  OIL-    product  t>  <he 
*511  bo.  t'.ic  amount  soKdit. 


AXNUIT:  Jb. 

E3CAMRJ 

1.  it  hi!  ar.nuity  of  TOJ.  e  5  years,  what  will 
he  diif  I'm-  ti;-               .  i  and  interest  at  the  end  of  said 

i:ig  computed  at  5  per  cent,  per 
••:ii  r  IT.     f.  s. 

iiitu-cit  of  7dl.  at  5  per  cent  for        1 —  3  10 

^:—  7    o 

3—10  10 
4—14     0 

£d.  And  5  yrs.  annuity.;.!  7 (:l.  per  jr.  is       350     0 

Jlns.  £385     o 

2.  A  lur;  iet  upon  a  lease  of  7  year.--,  at  400 
dollars  per  annum,  and  tlie  rent  beii:£  i:»  a.-re;tr  for  the 
whole  t«'fn:,  I  demand  the  suir.  due  at  the  end  of  the  term, 
simple  interest  being  allowed  at  6/.  [.erce'it  per  annum  ? 

SS304. 


To  find  tl.e  present  worth  of  an  annuity  at  simple  into:  i 
RULE. 

Find  (lie  present  v,  orth  of  each  year  by  itself,  discount  - 
iii^-  iVi  :u  i!ie  time  it  ialls  due,  and  the  sum  of  nil  tl 
jtresent  u-o:ti:s  will  be  the  }•-.  oci. 

EXAMl'LF.S. 

I.  V»'liaf.  is  the  present  vortli  of  400  dols.  per  .iiiiiiim. 
t->  fontinue  -I-  years,  at  t;  per  rent,  per  annum  ? 

5S49  =  Pres.  worth  of  1st  v,-. 


1  1  2      •  100  '  -  400  :  S57,l4285  =     --  2d  yr. 

L18f;  '538,98305=     -  3d  yr. 

1  24  j  322,58064  =     -  4th  yi  . 

Jns.  S1396,OG503=S1396,  Gets.   5m, 
2.  lieu-  much  present  money  is  equivalent  to  an  an- 
nuity of  100  dollars,  to  continue  3  years  j  rebaiti  being 
made  at  6  percent.  ?  .flus.  g268,  37c£s.  l»i. 

R.  N'.'hat  is  BO',  yearly  rent,  to  continue  5  year*,  worth 
in  ready  money,  at  (V.  per  cent.  ?    Ans.  £340  15s.  -f- 

12* 


1J8  EQUATION'   Or    1'AYMENTX 

EQUATION  OF  PAYMENTS, 

IS  finding  the  equated  time  to  pay  at  once,  se^ 
debt*  due  at  different  periods  of  tinu...  so  that  uo  lo.ss  shall 
be  sustained  by  either  party. 

RULE. 

Multiply  each  payment  by  it*  time,  and  divide  t! 
gf  the  several  products  by  the  \vholc  debt,  and  the  <ju!>: 
will  be  the  equated  time  for  the  payment  of  the  whole.  , 

XXAMl'LICS. 

1.  A  owes  B  3PO  dollars,  to  be  paid  as  follows — vi/.. 
100  dollars  in  6  months,  1*0  dollars  in  7  months,  and  160 
dollars  in  10  months :  What  is  the  equated  time  f.>. 
pxyinunt  of  the  whole  debt  ? 

100  x     G  =.     GOO 

120  x     7  =     840 

160  x  10  «=  1600 


580  )5040(8  months.   Jus. 

:l.  A.  merchant  hath  owing  him  3001.  to  be  paid  as  fol- 
:  .'>'<')/.  at.  ^  :.<,!)Mths,  lOOt.  at  5  months,  and  the  rest  at 
8  months  ;  and  it  i.->  agreed  to  make  one  payment  of  the 
\vhole  ;  I  demand  the  equated  time  ?     Jlns.  6  months. 

3.  F  owes  II  1000  dollars,  whereof  200  dollars  is  to  be 
puhi  400  dollars  at  5  months,  and  the  rest  at  15 

months, but  they  v™\  oc  to  make  one  payment  of  the  whole : 
J  demand  \  finie  must  be  ?       Jius.  8  month*. 

chant  lias  due  to  him  a  certain  sum  of  money, 
to  be  |  .i:«i  on  •  ;  months.  .ini>.  thir(fat3  months, 

.mi!  ;  is  j  A\lut  is  tiif  time  for 

;he  j  .  .ie  ?  jli'.s.  4\-  months. 

. 


BARTER, 

J  S  ti  •  of  one;  commodity  for  <i.'  •!  di- 

r»rts  i:.ercli;:iifs  and  traders  how  to  make  the 
<  T  j  uify. 

RULK. 

Vind    tlic  value  of  I';.-   (•'iinmodiLy  wl:'»:,e  <|iut. 
v,n  find  \\hat  ^'uintity  of  the  oilier  at  tht- 


BARTER.  ISf 

rate  can  be  bought  for  the  same  money,  and  it  gives 
the  answer.  / 

EXAMPLES. 

1.  "NVhat  quantity  of  flax  at  9  cts.  per  Ib.  must  be  givem 
in  barter  for  12lb.  of  indigo,  at  2  dols.  19  cts.  per  Ib.  ? 

12lb.  of  indigo  at  2  dols.  19  cts.  per  Ib.  comes  to  26 
dols.  28  cts.— therefore,  As  9  cts.  :  lib.  :  :  26,28  ets.  : 
292  the  answer. 

9.  How  much  wheat  at  1  dol.  25  cts.  a  bushel,  must  be 
given  in  barter  for  50  bushels  of  rye,  at  70  cts.  a  bushel  ? 

Ans.  28  bushels. 

5.  How  much  rice  at  28s.  per  cwt.  must  be  bartered 
for  34cwt.  of  raisins,  at  5d.  per  Ib.  ? 

Jlns.  Scict.  Sqrs.  9££f  Z&. 

4.  -How  much  tea  at  4s.  9d.  per  Ib.  must  be  given  in 
barter  for  78  gallons  of  brandy,  at  12s.  3^d.  per  gallon  ? 

Ans.  201  Ib.  13-ffoz. 

5.  A  and  B  bartered  :  A  had  8£cwt.  of  sugar  at  12  cts 
per  Ib.  for  which  B  gave  him  18cwt.  of  flour;  what\vas 
the  flour  rated  at  per  Ib.  ?  Ans.  5  bets. 

6.  B  delivered  3  hhcls.  of  brandy,  at  6s.  8d.  per  gallon, 
to  C,  for  126  yds.  of  cloth,  what  was  the  cloth  per  yard  ? 

.2ns.   10s. 

7.  D  gives  E  250  yards  of  drugget,  at  30  cts.  per  yd. 
for  3l91b.  of  pepper;  what  does  the  pepper  stand  him 
in  per  Ib.  ?  Ans.  QScts.  5Ty/rc. 

8.  A  and  B  bartered:  A  had  41cwt.  of  rice,  at  21s. 
per  cwt.  for  which  B  gave  him  20/.  in  money,  and  the 
rest  in  sugar,  at  8d.  per  Ib. ;  I  demand  how  much  sugar 

.  B  gave  A  besides  the  20J.  ?          Ans.  6cwt.  Oqr.  19J/6. 

9.  Tw.o  farmers  bartered :  A  had  120  bushels  of  wheat, 
at  1J  dols.  per  bushel,  for  which  B  gave  him  100  bushels 
of  barley,  v/orth  65  cts.  per  bushel,  and  the  balance  in  oats 
at  40  cts.  per  bushel ;  wnat  quantity  of  oats  did  A  receive 
from  B  ?  Ans.  287$  bushels. 

10.  A  hath  linen  cloth  worth  20d.  an  ell  ready  mone/; 
but  in  barter  he  will  have  2s.    B  hath  broadcloth  worth 
14s.  6d.  per  yard  ready  money,  at  what  price  ought  B  to 
rate  his  broadcloth  in  barter,  so  as  to  be  equivalent  te 
A's  bartering  price  ?  Ana.  17s.  4a. 


140  '    tOSS   AND    GAIN. 

11.  A  and  B  barter:  A  hath  145  gallons  of  brandy  at 
1  dol.  20  cts.  per  gallon  ready  money,  but  in  barter  he 
will  have  1  dol.  35  cts.  per  gallon :  B  lias  linen  at  58  cts. 
per  yard  ready  money;  how  must  B  sell  his  linen  per 
yard  in  proportion  to  A's  bartering  price,  and  !:  r»v  many 
yards  are  equal  to  A's  brandy  r 

*9ns.  Barter  price  of  B's  linen  is  CScls.  Qhn.  and  he 
must  give  A  SOOt/r/s.  for  his  brand>. 

12.  A  has  225  yds.  of  shalloon,  at  2s.  ready  money,  per 
yard,  which  he  barters  Avith  B  at  2s.  5d.  per  yard,  takn 
indijro  at  12s.  fid.  per  Ib.  which  is  worth  but  10s.  how 
Tiiucii  indigo  will  pay  for  the  shalloon;  and  who  gets  the 
best  bargain  ? 

.0//S.  43£/6.  at  barter  price  will  pay  for  the  shalloon, 
and  B  has  the  advantage  in  baiter. 

Value  of  A's  cloth  at  cash  price,  in  :     10 

Value  of45^&.  of  indigo,  at  10s.  per  Ib. 

B  get*  th  e  !  >e  >  t  ba  rga  M i  by     £  0     15 


LOSS  AND   GAIN, 

AS  a  rule  by  which  merchants  and  traders  discover  tl 
profit  or  loss  in  buying  and  gelling  their  goods  :  it  also  in- 
•>f.rur*s  them  how  to  rise  or  full  in  the.  price  of  tlwir  good-, 
so  as  to  gain  or  lose  so  much  per  cent,  or  otherv, , 
Questions  in  this  rule  are  answered  by  the  Rule  of  Three. 

1.  Bought  a  piece  of  cloth  containing  SS  yard-.   '<>r 
191  dols.  25  cts.  and  sold  the  same  at  "2  d,> 

yard;  what  is  the  profit  upon  the  wholf  pi- 

. 

2.  Bought  12*  cwt.   (>f  rice,  at  T.  dols.  45   c.<->. 

and  sold  it  again  at  4  cts.  a  pound  ;  what  was  the  whole 
gain?  Jlns.  g!2,  XTcl*.  5>n 

S.  Bought  11  cwt.  of  sugar,  at  f>id.  per  Hi.  but  <-.«»nld 
r.'it  sell  it  nrnn  for  ir\y  ihore  than  2/.  iCs.  per  cwt. :  did 
v  my  bargain  ?  .'/«.•..  L'Kt.f^Z  11s.  4d. 

4.  Bought  44  Ib*  of  tea  fo  .  rinlbr 

ii/.  lf>s.  M. ;  what  was  the  profit,  on  tu-Ji  pound  ? 


TOSS    ANT>     GAIN  141 

5.  Bought  a  hhd.  of  molasses  containing  119  gallons, 
f,i  52  cte.  per  gallon ;  paid  for  carting  the  same  1  dollar 
25  cents,  and  By  accident  9  gallons  leaked  out ;  at  what 
rate  must  I  sell  the  remainder  per  gallon,  to  gain  13  dol 
l^i's  in  the  whole  ?  Arts.  69cte.  2/n.-f 

II.  To  know  what  is  gained  or  lost  percent. 

RULE. 

Fir.*t  see  what  the  gain  or  loss  is  by  subtraction;  then 
As  the  price  it  cost  :  is  to  the  gain  or  loss  :  :  so  Is  100Z. 
01-  S1('0>  to  the  gain  or  loss  per  cent. 
KXAMPLES. 

1.  If  I  buy  Irish  linen  at  2s.  per  yard,  and  sell  it  again 
It  2<*.   fid.  per  vard  ;  what  do  I  gain  per  cent,  or  in  laying 
out  lOtif.?      As  :  2s.  8rf.  :  :  100J.  :  £33  6*.  8d.  Ans. 

2.  If  I  buy  broadcloth  at  3  dols.  44  cts.  per  yard,  and 
sell  it  again  at  4  dols.  30  cts.  per  yard  ;  what  do  I  gain 
per  cent,  or  in  laying  out  100  dollars  ? 

S  'cts.-} 

Sold  for  4,   301  g  cts.    cts.        $        $ 

Cost       3,   44  J>  As  3,  44  :  86  :  :  100  :  *? 

I  Jins.  25  per  cent, 

Gained  per  yd.     86J 

3.  If  I  buy  a  cwt.  of  cotton  for  34-doIs.  86  cts.  and  sell 
it  again  at  41  §  cts.  per  Ib.  what  do  I  gain  or  lose,  and 
what  per  cent.  ?    .  g  ots. 

I  cwt.  at  41$cts.  per  Ib.  comes  to  46,48 
Prime  cost  34,86 

Gained  in  the  gross,  $11,62 
As  34,36  :  11,62  ::  100  :  33|  Jlns.  Sty  per  cent. 

4.  Bought  sugar  at  8$d.  per  Ib.  and  sold  it  again  at  4l. 
iTs.  per  owt.  what  did  1  gam  per  cent.  ? 

Ms.  £25  19s.  5$d. 

5.  If  I  buy  12  hhds.  of  wine  for  204J.  and  sell  the  same 
again  at  14£.  17s.  6d.  perhhd.  do  I  gain  or  lose,  and  what 
per  cent.  ?.  Jlns.  I  lose  12 £  per  cent. 

6»  At  1  Jd.  profit  in  a  shilling,  how  much  per  cent.  ? 

Ans.     12  1»*. 


>.  ND     GA1.V. 


7.  At  £5  cts.  profit  in  a  dollar,  how  much  per  cent.  ? 

.ins.  25  per  cent. 

NOTE. — When  good-  are  bought  or  sold  on  credit,  you 
must  calctilui'  -count)  tlie  present  worth  of  their 

111  order  to  rind  your  tiue  gain  or  los?,  &c. 

KKAMPI.F.S. 

1    BousjM  164  yards  of  broadcloth,  at  14s.  6d.  per  yd. 
ready  money,  anil  ^r>!.i  i   1J4/.  10*.  on 

6  months  c'fdit:  what  did  I  £:iin  by  the  whole;  allow- 
ing discount  ar  • 

£•  ;   *• 

As  li'3  :  ICO  :  :   i^-t  10  :   loO     u  ,  orth. 

Gi   '  'T. 

2.  If  I  buy  rl.Mh  at  -  eight 

ms  creii/t.  . 

.  \hat  do  I  '  ii'i-  cent. 

.  cut. 


'Id,  to  gaiu 

RULK. 

:o  :  :  so  is   100/.  Or 
ti:e  profit  .'idcicii.  or  1  -.cted  :  to 

the  -  ,c. 


.  . 

1.  If  I  buy  Irisli  Hi.  ird  ;  now  must 

I  afll 

. 

I  buy  Rum  at  1  dn 

30  per  t 

3.  1  1  -  id  [tr 

As  Rl.:0  :  54cfs.   :  :  g 

'.ot  pvdving 
.  -  I  expected,  1  a;-  •  15  |j«'r 

"       J»;s.  14<.  K>i.-/. 


LOSS   AND    GAIN.  145 

5.  If  11  cwt.  1  qr.  25  Ib.  of  sugar  cost  126  doJs.  50  cts. 
how  must  it  be  sold  per  Ib.  to  gain  SO  per  cent.  ? 

Jus.  IZcis.  8m. 

6.  Bought  90  gallons  of  wine  at  1  dol.  20  cts.  per  gall, 
but  by  accident  10  gallons  leaked  out,  at  what  rate  must  I 
sell  the  remainder  per  gallon  to  gain  upon  the  whole  prime 
cost,  at  the  rate  of  12A  per  cent.  ?  •dns.  Sl>  olcfs.  8rrff»H. 

IV.  \Vhen  there  is  gained  or  lost  per  cent,  to  know 
what  the  commodity  cost. 

RULE. 

As  100/.  or  lOOdols.  with  the  gain  per  cent,  added,  or 
loss  per  cent,  subtracted,  is  to  the  price ;  so  is  100  to  the 
prime  cost. 

EXAMPLES. 

1.  If  a  yard  of  cloHi  be  sold  at  14s.  7d.  and  there  is 
gained  161.  I3s.  4d.  per  cent.  5  what  did  the  yard  cost  ? 

£.    *.     d.     s.  rf.          £. 
As    116  13  4  :  14  7  :  :  100  to  12s.  6<f.  Jns. 

2.  By  selling  broadcloth  at  3  dols.  25  r.ts.  per  yard,  I 
lose  at  the  rate  of  20  per  cent. ;  what  is  the  prime  cost  of 
said  cloth  per  yard?  Jlns.  g4,  06cfs.  Sim. 

3.  If  40  Ib.  (  f  chocolate  be  sold  at  25  cts.  per  Ib.  and  I 
gain  9  per  cent. ;  what  did  the  whole  cost  me  ? 

Jlns.  S9,  17r; 

4.  Bought  5  cwt.  of  sujjar.'  and  sold  it  again  at  1°  cents 
per  Ib.  by  which  I  gaineu  at  the   rate 

what  did  the  sugar  cost  me  per  cwt. 

h  S10,  TQcis.  '}m.+ 

V.  If  by  «  en  rate  there  is  so  mucli 
gained  or  lost  per  cent  ,1  !  01 

lost  per  cent,  ii'sald  at  another  rat-i. 

LE. 

As  the  first  price  :  is  to  '  >  <Vt.s.  with  th> 

per  ceru.  added,  or 
other  p:ice  :  t  -  or  loss  per  cei 

N.  K.  If  your  anbsver  e.Kceed  1001.  o,   i;K)  do! 
«xce.ss  is  your  gain  per  cent,  j  but  if  it  be  L 
U'ss  per  cent. 


144  FELLOWSHIP. 

EXAMPLES. 

1.  If  I  sell  cloth  at  5s.  per  yd.  and  thereby  gain  15  per 
cent,  what  shall  I  gain  per  cent,  if  I  sell  it  at  6s.  per  yard  ? 

s.    £.     8:   £. 

As  5  :  115  :  :  6  :  138  Jlns.  gained  38  percent. 

2.  If  I  retail  rum  at  1  dollar  .'>0  cents  per  gallon  and 
thereby  gain  25  per  cent,  what  shall  I  gain  or  lose  per 
cent,  if  I  sell  it  at  1  dol.  8cts.  per  gallon  ? 

8  cfs.    8        g  cts.    8 

1,5(1  :  125  :  :  1,08  :  90  Ans.  I  shall  lose  10  per  cent. 

5.  If  I  sell  a  cwt.  of  sugar  for  8  dollars,  and  thereby 
lose  12  per  cent,  what  shall  I  gain  or  lose  per  cent,  if  I 
sell  4  cwt.  of  the  same  sugar  for  36  dollars  . 

Jlns.  Hose  only  1  per  cent. 

4.  I  sold  a  watch  for  IT/.  Is.  5d.  and  by  so  doing  lost 
15  per  cent,  whereas  I  ought  in  trailing  to  have  cleared 
20  per  cent. ;  how  much  was  it  sold  under  its  real  value"? 

£•     £•*•«*•       £•     fr** 

As  85  :  17  1  5  :  :  100  :  20  1  8  the  prime  cost 
100  :  20  1  8  :  :  120  :  24  i  0  the  real  value. 
Sold  for    17  1  5 

£707  Answer. 


FELLOWSHIP, 

IS  a  rule  by  which  the  accompts  of  several  mcrt.har  • 
other  persona,  trading  in  partnership,  are  so  adj.; 
that  cadi  may  ha'.  ••  ihe  gain,  or  sustain  hit 

share  of  the   los«.  in  proportion  to  his  share  of  the 
stock.  —  Also  by  this  Rule  a  Ixmkriipt's  estate  may  !.- 


vided  among  hi^  creditors. 


SIMJLK 

Is  when  the  s*  •         -»ti  H 

trade  an  equal  term 

BUJ 

As  the  whole  stocl<  vhoie  gain  or  lost:  «o  i» 

tach  man's  particular  ktock,  to  liit>  pu  ticui*r  shtrewf  tH« 
or  low. 


JILtOWSHlP. 


PROOF. — Add  all  the  particular  shares  of  the  gain  or 
loss  together,  and  if  it  be  right,  the -sum  will  bo  equal  to 
the  whole  gain  or  loss. 


EXAMPLES. 


1.  Two  partucrs,  A  and  B,  join  their  stock  and  buy  a 
quantity  of  merchandize,  to  the  amount  of  820  dollars ; 
in  the  purchase  of  which  A  laid  out  S50  dollars,  and  B 
470  dollars ;  the  commodity  bein*  sold,  they  find  their 
clear  gain  amounts  to  250  dols.     What  is  each  person's 
share  of  the  gain  ? 

A  put  in  350 

B 470  * 

A*  a°o  -2™--  5550  :  106,7073  +  A's  share. 
•  £  470:  143,3920+ ITs  share. 

Proof  249,9999 -{-  «=g£50 

2.  Three  merchants  make  a  joint  stock  of  1 200/.  of 
which  A  put  in  240  J.  B  360/.  and"C  60G/.— and  by  trading- 
they  gain  325J.  what  is  each  one's  part  of  the  gain  ? 

Ans.A>spart£65.  /Ts  £97  105.  C'«£162  10s. 

3.  Three  partners,  A,  B,  and  C,  shipped  108  mules  for 
the  West-Indies;  of  which  A  owned  48,  BS6,andC  24. 
But  in  stress  of  weather  the  mariners  were  obliged  to 
throw  45  of  them  overboard ;  I  demand  how  much  of  tho 
loss  each  owner  must  sustain  ? 

Jus.  A  20,  B  15,  and  C  10. 

4.  Four  men  traded  with  a  stock  of  800  dollars,  by 
which  they  gained  307  dols. -A 's  stock  was  140  dols.  B's 
260  dols.  C*s  300  dols.  I  demand  D's  stock  and  what 
each  man  gained  by  trading  ? 

Jlns.  D's  stock  was$lQV,and  Ji  gained  g53,  72c£s.  5m. 
B  899,  77ic«s.  6yjgll5,  lZ}cts.  and  D  S38,  STicts. 

5.  A  bankrupt  is  indebted  to  A  21  ll.  to  B  3(Ktf.  and  to 
C  391/.  and  his  whole  estate  amounts  only  to  6751.  10s. 
which  he  gives  up  to  these  creditors;  how  MUCH  must 
each  have  in  proportion  to  his  debt  ? 

Jlns.  Ji  must  have  £158  0*.  3|rf.  B  £224  13s.  *j(*   ana 
C  £392  16s   3|rf 

13 


148  COMPOUND    FELLO.WSIUJ?. 

6.  A  captain,  mate  and  £0  seamen,  took  a  prize  wdrth 
5501  dols.  of  which  the  captain  takes  11  shares,  and  the 
mate  5  shares  ;  the  remainder  of  the  prize  is  equally  di- 
vided among  the  sailors ;  how  much  did  each  man  re- 
ceive ?  $    cts. 

•ins.  The  captain  received,     1069,  75 
The  mate  486,  25 

Each  sailor  97,  25 

7.  Divide  the  number  of  350  into  5  parts,  which  shall 
be  to  each  other  as  2. 3,  and  4.    Jlns.  80,  120,  and  160. 

8.  Two  met  chants  have  gained  450^.  of  which  A  is  t* 
have  3  times  as  much  as  B  ;  how  much  is  each  to  have  ? 

Jlns.  .1  £337  10s.  and  B  /:  112  10s.— l-fS=l  : 
450  :  :  3  :  £337  10.>.  J's  .• 

9.  Three  persons  ate  to  share  GiXtf.  A  is  to  hav;i  •- 
tain  sum,  lias  much  again  :.- 

much  as  B.     1  demand  cacli  man's  part  ? 

.tois.  J  £<;»>.<;.,   V  £l.:3-£.  unl  ' 

10.  A  nnd  B  tiado'.!  t»>^cther  arid  ."aiiunl  100  d( 
:;  t;-!0  dols.  B  |'iit  in 

...nd  B's  stock? 

11.  A,  !?,  and  C,tra«led  \ 

>>do!-;.  ;tiid  ('  put  i>>   1:20  ydi;.  of  < 

il<  I  -.  uf  which  (..'•  ti.«ik  Kid  dn 
;  in  :  how  did  C  value  his  cloili  per  v-a: 
jrk.  ;u:'l  \v! i at  was  A  and  B's  part  of  i 
.   C  }>ut  hi  Hie  clutk  at  . 
R46,  G7</s.  (j/fi.-f  und  U  &8S,  SScfi 


COMPOUND  FELLOWSHIP, 

V^R  Fellowship  v. iili    t'.me,   is  occasioned   by  sf 
s'nai  v.ersbeiiig  continued  ii: 

U-nn  ot  time. 

RUtB. 

Alui',  •  ."..harc  i»v  the  ii:i;c  it  \'a» 

']5 

Is  lo  the  whole  i-ruu  cr  ' 

So  ;  .  IN? eh  iv;  i  uct, 

To  :  i  -'is-"1  w*' 


nsrnr.  147 

KXAMP1RS. 

1.  A,  B  ;md  C  hold  a  pasture  m  common,  fur  n'lich 
'hey  pay  19/.  per  annum.  A  put  in  8  oxen  tor  G  weeks  ; 
I',  12  oxen  lor  8  weeks;  and  C  12  oxen  for  12  weeks; 
what  must  each  pay  of  the  rent  ? 

£.     s.    d. 

8X   fi*=  481  f  43  :  ;i     3     4  A's  pt. 

8=  PC,  j  ,  :  6     6     8  IVs  — 

I-2XV.  -1  U  '->As  288  :  19/.  :  :--.   144  :  9  10     0  C's  — 


gj  -oof  19  0     0 

:  •'.'••  A  put   in  215 

ih>!s.  f>r  •  ':•  .ninulis,  but  by 

'oitune  t'u-v  lose  xl1  d  dols.  :  1-,  MV  nvist  they  nhavotiiC 

.  'a  SI  46,  2 

3.  Three  persons  liral  received  CQ5  dr.ls.  interest:  A 
iiacl  put  in.-'(;f*0  do's,  for  12  mouths,  B  3000  dels.  f>,r  15 
months,  and  C  :  ho\v  much  is  each 

man's  part  of  .  t  ? 

i,  5  &£25  and  C  S200  ^ 

-  1  ;   bv  trading;  110/.  12s.:    Ar* 

ock  w.?-  ths,  and  F/s  200Z.  for  6i 

months;  \v',i;r  MI'S  p.irt  of  the  p;ain  ? 

*Jw«.  *iJ5j  yjxs  /;60  l;ls.  8  ]  ^..^V 

J.  TV.  o  men  Jiants  CI.UT  into  pnrtnersliip  for  18  months. 
A  at  first  put  iut'>  stock  500  dollars,  and  at  the  end  of  S 
months  he  put  i;i  100  dollars  more  ;  li  at  first  put  in  800 
dollars,  and  at  4  month's  end  took  out  200  dols.  At  the 
expiration  of  the  time  they  find  they  have  gained  700  dol- 
lars ;  what  is  each  man's  share  of  the  gain  ? 

$8324.  07  4-M's  share. 


92  5+JSPs.      do 

6.  A  and  15  companied  ;  A  put  in  the  first  of  January, 
1000  dols.  ;  but  B  could  not  put  in  any  till  th«  first  of 
May:  what  did  he  then  put  in  to  have  an  equal  share 
.  A  at  the  year's  end  ? 

J/o.       g          Jjb.  S 

<LJ  12  :  1000  :  :  8  :  1000x12=1500 

8 


l^S  .  .1LE    KCLE    OF   THREE. 

DOUBLE  RULE  OF  THREE. 

-i  HE  Double  Rule  of  Three  teaches  to  resolve  at 
ouce  such  questions  as  require  two  or  Diore  statings  in 
simple  proportion,  whether  direct  or  inverse. 

In  thi.s  rule  t'u-re  iiro.  always  fi.-e  terms  given  to  find  a 
sixth;  the  three  first  terms  of  which  are  a  supposition, 
the  two  !  .;t  a  demand. 

RULE. 

In  string  the  question,  place  the  terms  of  tlie  supposi- 
tion so  that  the  principal  cause  of  loss,  gain  or  action  pos- 
sess the  ilrst  place:  that  which  signifies  time,  distance  of 
place,  itc.  in  the  second  place  5  and  the  remaining  tenn 
in  the  third   place.     1'lac.e  the  terms  <>!'  demand,  under 
;  of  the  same  kind  in  the  supposition.     If  the  blank 
place  or  i         »      ,ut,  fail  under  the  third  term,  the  pro- 
portion  is  direct;  then  multiply  the  first  and   second 
.s  together  for  a  divisor,  and  the  other  three  for  a 
lend  :  but  if  the  blank  fall  under  the  first  or  second 
•  proportion   is  inverse  ;  then  multiply  the  third 
i  terms  together  for  a  divisor,  and  th.e  other 
three  for  a  dividend,  and  the  quotient  will  be  the  answer. 

EXAMPLES. 

1.  If  7  men  can  Iniilit  S6  roUs  of  wall  in  5  tkysj  liow 
rn.-ny  rods  can  2u  aien  build  in  14  days? 

:  36  Terms  of  supposition. 
Terms  wf  demand. 


7x^=^)10080(480  wda  Jns,. 
2.  If  101  ,ain  6/.  interest  in  12  months, 


•  in  7  mo: 


:  U'7/4o.  :  :  6/.  Int. 
400    :    7  Ana.  14L 


GOXJOINKD     PROrORTIOV.  149 

S.  If  100/.   will  gain  61.  a  year  ;  in  what  time  will 
400/.  gain  14/.         £.       mo.        £. 
100  :  12  :  :     6 
400  :         :  :  14         Jlns.  7  iiWHf/&. 

4.  If  400/.  gain  14L  in  7  months ;  what  is  the  rate  per 
otat.  per  annum  P        £.    wo.     Int. 

400  :  7  :  :  14 

100  :  12  Jlns.  .£6. 

5.  "What  Principal  at  GJ.  per  cent,  per  annum,  will  gain 
141.  in  7 months,?  mo.      Int. 

100  :  12  :  :    6 

7  :  :  14  Jn*.  £400. 

6.  An  usurer  put  out  861.  to  receive  interest  for  the 
game  :  and  u^on  it  had  continued  8  months,  he  received 
principal  and  interest,  881.  17s.  4d.;  1  demand  at  what 
rate  per  ce?if.  pei-aiui.  he  received  interest  ?  Jlns.  5  per  ct. 

7.  If  20  bushels  of  wheat  are  sufficient  for  a  fUisily  of 
8  persons  5  months,  how  much  will  be  sufficient  for  4  per- 
sons 12  months  ?  dns.  24  bushels. 

8.  If  50  men  perform  a  piece  of  work  in  20  days  ;  how 
many  men  will  accomplish  another  piece  of  work  4  times 
83  large  in  a  fifth  part  of  the  time  ? 

30  :  20  :  :  1 

4  :  :  4  Jns.  600. 

9.  If  the  earriage  of  5  cwt.  5  qrs.  150  miles,  cost  24 
dollars  58  cents  ;  what  must  be  paid  for  the  carriage  of 
7  cwt.  2  qrs.  25  Ib.  64  miles  at  the  same  rate  ? 

Jlns.  g14,  08c£s.  G??i.-f 

10.  If  8  men  can  build  a  wall  20  feet  long,  6  feet  high 
and  4  feet  thick,  in  12  days  ;  in  what,  time  will  24  men 
build  one  200  f«et  long,  8  feet  high,  and  6  f*et  thick  ? 

8  :  12  :  :  20x6x4 


S4  :  200x8x6     80  days, 


CONJOINED  PROPORTION. 

IS  wliei'  the  coins,  weights  or  measures  of  several  coun- 
tries are  compared  in  the  same  question;  or  it  is  joi-iing 
proportions  together,  and  by  t)>«  relation  which 
13* 


i50  CONJOINED     PROPORTION. 

several*  antecedents  have  to  their  consequents,  the  pro- 
portion between  the  first  antecedent  ano  the  last  conse- 
ouent  is  discovered,  us  well  as  the  proportion  between 
Uie  others  in  their  several  respects. 

NOTE. — This  rule  may  generally  be  abridged  by  can- 
celling equal  quantities,  or  terms  that  happen  to  be  the 
same  in  both  columns :  and  it  may  be  proved  by  as  many 
statings  in  the  Single  Rule  of  Three,  as  the  nature  of  the 
question  may  require. 

CASE  I. 

When  it  is  required  to  find  how  many  of  the  first  sort 
of  coin,  weight  or  measure,  mentioned  in  the  question, 
arc  equal  to  a  given  quantity  of  the  last. 
RULE. 

Place  the  numbers  alternately,  beginning  at  the  lei;. 
hand,  and  let  the  last  nnmlvr  stand  on  the  left  hand  cot 
umn ;  then  multiply  the  left  hand  column  continually  for 
a  dividend,  and  the  right  hand  for  a  divisor,  and  the  quo- 
tient will  be  the  ui;:-;wt>r. 

EXAMPLES. 

1.  If  lOOlb.   English  make  »5lb.  Flemish,  ami    19lb- 
Flemish  how  m;ir:y  pounds  English 
are  equal  to  50lb.  at  Bologna  ? 

Ib.  Ib. 

100  Eng.=95  Flemish. 
ID  Fie.  ^=£5  Bologna. 
50  Bologna.  Then  95x2.5  =23 T5  the  dh 

95000  dividend,  and   2375)95000(40  .. 

2.  If  4Glb.  at  New- York,  make  48lb.  at  Antwerp,  and 
SOlb.  at  Antwerp,  make  SGlb.  at  Leghorn ;  how  many 
ih.  at  New-York  are  eqnal  to  14411).  .  :i  ? 

V/HS.  lOOlb. 

3.  If  70  braces  at  Venice  be  equal  to  75  braces  at  Leg- 
horn, and  "  braces  at  Leghorn  be  equal  io  4  Amor- 
yards;  ho.v  many  braces  at  N'enicc  arc  eciu;il  io  (J4  Ame- 

•  lt)4A 
CASE  II. 

When  it  is  required  to  find  how  many  of  the  last  sort 
of  coin,  weight  or  measure,  mentioned  in  the  question 
are  equal  to  a  given  quantify  of  the  first. 


EXCHANGE.  15 

RULE; 

Place  the  numbers  alternately?  beginning  at  the  left 
hand,  and  let  the  last  number  stand  on  the  right  hand; 
then  multiply  the  fii'st  row  for  a  divisor,  and  the  second 
for  a  dividend. 

EXAMPLES. 

1.  If  24lb.  $  New -London  inake  20!b.  at  Amsterdam, 
and  r>'>:\j.  at  Amsterdam  ';')!!>.  at    Pni'is ;  how  manylat 
Paris  are  equal  to  40  at  New-London  ? 
L',"t.     Right. 
24  ==  2ft        20  x  60  X  40  =  48000 

50  =  GO  =  40  Ans. 

40         24  x  50     =  1200 

'2.  If  501  b.  at  New-York  make  45  at  Amsterdam,  ami 
80!b.  at  Amsterdam  snake  103  at  Dantzic ;  how  many  ft. 
at  Dant/.ic  arc  equal  to  240  'it  N.  York  ?  .2ns.  278-^ 

3.  Jf  W  bi-aces  at  Lowborn  be  equal  to  11  vares  at 
Lisbon,  and  40  varcs  at  Lisbon  to  80.  braces  at  Lucca; 
how  many  braces  at  Lucca  are  ^qiial  to  100 'braces  at 
Leghorn  r  Ans.  110 


EXCHANGE. 

_L» Y  tin's  rule  merchant  'iat  sum  of  money  ou^ht 

to  l>e  received  in  one  country,  for  any  sum  of  different 
specie  paid  in/fcaother,  according  to  tlie  given  course  of 
exchange. 

To  reduce  the  monies  of  foreign  nations  to  tRat  of  tht 
United  States,  you  may  fons;;.lt  the  follov;ing 

TAli 
Showing  the  value  or  the  monies  of  account,  of  foreign 

nations,  estimated  in  Federal  Money.*    g  cts. 
Pound  Sterling  ot  Great-Britain,  4  44 

Pound  Sterling  of  Ireland,  4  10 

Lpre  of  France,  "  18i 

Guilder  or  Florin  of  tie  U.  Netherlands,          u  .39 
Mark  Banco  of  Kmnburglv,  <>  351 

Rix  Dollai-  of  Denmark,  1     0 

*Laws  U.  3.  Jt.j 


152  EKCH.AX0K. 

Rial  Plate  of  Spain,  0  1» 

Milrea  of  Portugal,  t  24 

Tale  of  China.  1  48 

Pagoda  of  India,  1  94 

Rupee  of  Bengal,  0  55  J 

I.  OF  GREAT  BRITAIN. 

EXAMl'I.KS. 

1.  In  45/.  10s.  sterling,  how  many  dollars  and  ». 

A  pound  sterling  being^444  cents 
Therefore-—  As  I/.  :  444cfs.  :  :  45,5/.  :  20202<*fs.  Jlns. 

2.  In  500  dollars  how  many  pounds  sterling? 

As  444cte.  :!/.::  50000et*.  :  11&.  12s.  5</.-f  J 


II.  OF  IRELAND. 

EXAMPLES. 

1.  In  90f.  IDs.  Gd.  Irish  money,  how  many  cents  ? 

M  Irish  =»4l6c/s. 

£.    c/.s.          /;.  cfs.  R     c/s. 

Therefore—  As    1  :  410  :  :  90,525  :  S71151=5fl,  15i 

2.  In  1G8  dols.  10  cts.  how  many  pounds  Irish  : 
As410c/s.  :  M.  :  :  l6810cte.  :      41  Irish.     Jns. 


III.  OF  FRANCE. 

Accounts  are  kept  in  li\res,  sols  arid  deraers. 
5  12  dcniers,  or  pence,  make  1  sol,  or  sliiliing, 
^  20  sob,  or  shillings,       —     1  livre,  or  pound. 

EXAMPLES. 

1.  In  250  livrcs.  8  ?o!.i,  how  many  doll;  -s  and  cents? 

1  livie  of  France  =18$  cts.  "or  185  mills. 
£.     i».  £.  in.         g.  c/s/w. 

As  1  :  1S5  :  :  3fiu.4  :  4^,3-24=46,  3  si   4     jJns. 

2.  Rttduce  87  dol.^.  4.'i  rts.  7  in.  into  livres  of  France. 
milts,     lie.       mills,      liv.    sn.  den. 

As  185  :  1  :  :  87457  :  472  14  9-f-  Jlns. 

IV.  OF  THE  U.  NETHERLANDS. 

Accounts  are  kept  here  in  guilders,  stivers,  groats  and 
phennin^s. 

3  phenmngs  make  1  jj;roat. 
'2  g  .  —       1  stiver. 

^0sti.-  —       1  guilder,  or  florin. 

A  gui'der  is—  ."9  cents,  or  590  mills. 


5-IB 


EXCHANGE.  153 

EXAMJ-; 
Reduce  124  guilders,  14  stivers,  into  federal  money* 

On  I  ft     d-  c.  m. 

As  1    :   *9  :  :   i,:4.r  :  48,    633    Jlns. 

miU*.     6r.       mills:      G. 
As  3SO  :  1  :  ;  48633  :  124,7  Proof. 

V.  OF  HAMBURGH,  IN  GERMANY 

Accounts  are  kept  in  Hamburgh  in  marks,  sous  anil 
denieis-lubfc,  awl  by  some  in  rix  dollars. 

'       deniers-lubs  make  1  sous-lubs. 
16  sous-hibs,        —     1  indYk-lubs. 
3  raavk-lubs,       —     1  rix-dollar. 
No  IK. — A  mark  is  =  334-  cts.  or  just  •$  of^  dollar. 

RULE. 

()jvid<pthe  marks  by  3,  the  quotient  will  be  dollars. 

IV.vAMPLES. 

Reduce  641  marks,  8  sous,  to  federal  moncv. 
3)641,5 

g21S,835  Jlns. 

But  to  rc:iuce  Federal  Money  into  Marks,  multiply 
the  given  sum  by  3,  &c. 

EXAMPLES. 

iteduce  121  dollars.  90  cts.  into  marks  banco. 
121,90 


3G5,70=S65  marks  11  sous,  2,4  deu.  Jlns. 
VI.  OF  SPAIN. 

Acconnts  are  kept  iti  Spain  in  piastres,  rials  and  marvadies. 

$34  marvadius  of  plate  make  1  rial  of  plate. 

2  8  rials  of  plate  —  1  piastre  ci  piece  of  8. 

To  reduce  rials  of  plate  to  Federal  Money. 

Since  a  rial  of  plate  is  =  TO  cents,  or  1  uime,you  need 
only  call  the  rials  so  many  dimes,  and  it  is  done. 

EXAMPLES. 

455  riale=485  dimes,=»48  dols.  50  cts.  &c. 


154  EXCHANGE 

Biif  f  o  mince  cwits  into  rials  of  plate,  divide  by  10 — 

Thus,  845  c«its-r-l  0=84,5=84  rials,  17  marvadies,  &c. 

VII.  OF  PORTUGAL. 

Accounts  are  kept  throughout  this  kingdom  in  milreas. 
and  reas,  reckoning  1000  reas  to  a  milrca. 

NOTK. — A  vnilrea  is  *=  124  cents  ;  therefore,  to  reduce 
inilreas  into  Federal   Money,  multiply  by  124,  ain: 
product  will  be  cents,  and  decimals  of  a  cent. 

EXAMPLES. 

1.  In  540  milreas  how  many  cents  ? 

340x124=42160  cents,=£421,  GQcts.  Ans. 

2.  In  'II 1  milreas,  48  reas,  how  many  cents  ? 

NOTK. — When  the  reas  are  less  than  100,  place  a  cy- 
pher before  them.— Thus,  21 1.048x124=26169,952  cts. 
or  201  dol*.  (i9  cents,  9  mills. -f  dns. 

But  to  reduce  cents  into  milreas.  divide  them  by  124  ; 
and  if  decimals  at  is£,  you  must  carry  on  the  quotient  as 
far  as  three  decimal  places ;  then  the  whole  numbeis 
thereof  will  be  the  milreas,  and  the  decimals  will  be  th* 
reas. 

EXAMPLES. 

1.  In  4195  cents,  how  many  milreas  ? 

4 195-f- 124=33,850+  or  SSmi/r.  830r«rs.  Jlns. 

2.  In  24  dels.  92  cts. how  many  milreas  of  Portugal? 

•0ns.  iiO  mil  reas,  096  reas. 

VIII.  EAST  INDIA  MONEY. 

To  reduce  India  Money  to  Federal,  vi/,. 
f  Tales  of  China,  multiply  with  148 
<  Pagodas  of  India,  194 

(^  Rupee  oi  Uerigal,  55 i 

EXAMPLKS. 

J.  In  €41  Tales  of  China,  how  many  cents? 

Jhis.  9486* 
2.  In  50  Pagodas  of  India,  how  many  cents  ? 

j/is.  oroo 

5.  la  98  Rupees  of  Bengal,  IKSVT  many  cents  ? 

Jin*.  54.W 


•  VULQAR    FRACTIONS.  1  5p 

VULGAR  FRACTIONS 

llAVING  briefiy  introduced  Vulgar  Fractions  iirune- 
(1  lately  after  reduction  of  whole  numbers,-  and  £> 
general  definition^,  and  a  few  such  problems  therein  as 
were  necessary  to  prepare  and  lead  the  scholar  r.umedi- 
:-.i.elv  to  decimals:  the  4earner  is  therefore  requested  to 
read  those  general  definitions  in  page  74. 

Vulgar  Fractions  are  eitiver  proper,  improper,  single-. 
compound,  or  mixed. 

1.  A  single,  simple,  or  proper  fraction,  is  when  the  nu- 
merator is  less  than  the  denominator,  as  4  $  f  4-|?  &c. 

•2.  An  Iwpvoipstv  Fraction,  is  when  the  numerator  cx- 
.     .Icnoimnator,  as  -|  %•  L2,  occ. 

.'j.  A  Cumpou'itd  Fraction,  is  the  fraction  of  a  fraction, 
«(;ii:ned  bv  the  \vord  of,  thus.  f  of  ~jl2,  A  of  §  of  3,  &c. 

4.  A  JJi.vcd  Nuinber.  is  composed  of  a  whole  r.  umber 
and  a  fracti.iii,  lluis,  'Si,  14/3-,  i^c, 

5.  Any  whole  number  may  be  expressed  like  a  fraction 

.  line  under  it.  and  puKing  1  for  cienoinina- 


. 
thus,  8=%,  and  1;"  &c. 

he  coiWncu  measure  of  t;vo  or  mare  numbers,  is 
that   number  \\hich    will  divide  each  ol   thi-.m  witl^mt  .1 
."imler;  thus,  6  is  thecoi»mun  measure  of  H,  ^4-jn(l 
.-  which  will  do  this,  is  called 
Lie  greatest  common  measure. 

r.  A  number,  which  can  be  measured  bv  two  or  more 
numbers,  is  called  their  com-iiwH  inulfipie  :  and  if  it  l< 
least  Uttmber  that  can  be  so  measured,  it  is  called  the  I-  •••'t 
''"uunun  multiple:  tlius,  24  is  the  common  multiple  of  ^ 
5  and  4  _;  but  their  least  common  multiple  i-, 

To  (ind  tl)e  least  common  multiple  of  two  or  more 
numbers. 

RULE. 

1.  IVivi'lc  bv  any  number  that  will  divkl*  hvo  or 
of  the  a,ivci»  nuaiijers  w;tiifiut  a  rtnutim'uT,  r.;' 
(jiioticnts,  tugetlier  with  the  undivided  numbers. 
beneath. 

2.  Divide  the  second   lines  ag  before,  an: 
tb'.c  aie  no  Ivy  mmmers  1h«t  caa  be  divided  ; 


156      REDUCTION  OK  VULGAR  FRACTIONS. 

continued  product  of  the  divisors  and  quotients,  will  give 
the  multiple  required. 

EXAMPLES. 

1.  What  is  the  least  common  multiple  of  4, 5,  6  and  10  ? 
Operation,         x5)4    5    6    10 


X2)4    162 
X2    1x3       I 

5  X2x2x3=»6p   .tow. 

'.  What  is  the  least  common  multiple  of  6  and  8  ? 

..ins.  24 

3.  What  is  the  least  number  that  3,  5,  8  and  12  will 
measure?  Jins.   120 

4.  What  is  the  least  number  that  can  be  divided  by  the 
t)  digits  separately,  without  a  remainder  ?    Jkis.  2520 

REDUCTION  OF  VULGAR  FRACTIONS, 

IS  the  bringing  them  out  of  one  form  uito  another,  in 
order  to  prepare  them  for  the  operation  of  Addition,  Sub- 
traction, &c. 

CASE  I. 

To  abbreviate  or  reduce  fractions  to  their  lowest  terms. 
RULE. 

1.  Find  a  common  measure,  by  dividing  the  greater 
term  by  the  less,  and  this  divisor. by  the  remainder,  and 
so  on,  always  dividing  the  last  divisor  by  the  last  remain- 
der, till  nothing  remains,  the  last  divisor  is  the  common 
measure.* 

2.  Divide  both  of  the  term1*  of  the  fraction  by  the  com- 
mon measure,  and  the  quotients  will  make  the  fraction 
required. 

*  To  find  the  greatest,  cmnnwn  measure  uf  jnorf  than 
t  ••!'')  numbers,  you  must  find  the  greatest  common  measure 
of  two  oftlu-masper  rule  abnvr  ;  then,  nf  that  commqn 
measure  and  out  of  the  other  numbers,  and  so  on  through 
all  tiw  numbers  to  the  ltr«t  $  then  will  tke  greatest  common 
measure  last  found  be  tfie  answer. 


OF   VULGAR   FRACTIOUS.  157 

OR,  If  you  choose,  you  may  take  that  easy  method  in 
Problem  I.  (page  74.} 

EXAMPLES. 

Jr.  Reduce  $$•  to  its  lowest  terms. 

48)||(i  Operation. 

£fv*i(6  common  mea.  S)?£=f  Ans. 

J^   Bern. 

2.  Reduce  £f  to  its  lowest  terms.  3ns.  £5 

3.  Reduce  |ff-  to  its  lowest  terms.  fins.  |£ 

4.  Reduce  f£||  to  its  lowest  terms.  3ns.  $ 

CASE  II. 

T§  reduce  a  mixed  number  to  its  equivalent  improper 
fraction. 

RULE. 

Multiply  the  whole  number  by  the  denominator  of  the 
tiven  fraction,  and  to  the  product  add  the  numerator. 
this  sum  written  above  the  denominator  will  form 
fraction  required. 

EXAMPLES. 

1.  Reduce  45|  to  its  equivalent  improper  fraction. 

45x8+7  =  -' 

2.  Reduce  19}  J  to  its  equivalent  improper  frsction. 

3ns,  V/ 

5.  Reduce  iCvj^  to  an  improper  fraction. 

3ns.  VA* 
4.  Reduce  Cmj  to  its  equivalent  improper  fraction 

•J.,0        2  tO  Si 

••"*'       Ti  » 

CASK  III. 

To  find  the  value  of  an  improper  fraction. 
RULE. 

Divide  the  numerator  by  the  denominator,  .•»•• 
quotient  will  be  the  value  sought. 

EXAMPLES. 

1.  Find  the  value  of  V  .   5)4,- 


2.  Find  the  value  uf 

3.  Find  the  v;il'M 

4.  Find  the  value  of«  ^V  s  .  fii^jrj 
*.  Find  the  va4ue  of  V 


158  REDUCTION  OF   VULGAR  FRACTIOUS. 

CASE  IV. 

T«  reduce  a  whole  number  to  an  equivalent  fraction,  hav- 
ing a  given  denominator. 

RULE. 

Multiply  the  whole  number  by  the  given  denominator ; 
place  the  product  over  the  said  denominator,  and  it  will 
form  the  traction  required. 

EXAMPLES. 

1.  Reduce  7  to  a  fraction  whose  denominator  will  be 
9.  Thus,  7x9=63,  and  6,/  ike  answer. 

2.  Reduce  18  to  a  fraction  whose  denominator  shall 
be  12.  Jns.  2TV6 

3.  Reduce  100  to  its  equivalent  fraction,  having  90 
for  a  denominator.  .0ns. 


CASE  V. 

To  reduce  a  compound  fraction  to  a  simple  one  of  equal 

value. 

RULE. 

1.  Reduce  all  whole  and  mixed  numbers  to  their  equi 
valeut  fractions. 

2.  Multiply  all  the  numerators  together  for  a  new  nit 
merator,  and  all  the  denominators  ior  a  new  denomint 
tor;  and  they  will  form  the  fraction  required* 

EXAMPLES. 

1.  Reduce  £  of  f  of  £  of -^  to  a  simple  question. 

1X2X3X4 

=^=— rV  •*»«• 

2X5X4X10 

2.  Reduce  4  of  £  of  jto  a  single  fraction.    Jlns.  £/f 

3.  Reduce  £  of  |£  of  ^  to  a  single  fraction. 


4.  Reduce  $  of  |  of  8  to  a  simple  fraction. 

•Jlns.  VV 
4-  Reduce  }  «f  ^|  of  42}  to  a  simple  fraction. 


NOTE.  —  If  ine  denominator  of  any  member  of  a  com- 
p4uml  fraction  b«  «qi*J  to  ti<«  cuiuerator  of  another 


159 

member  thereof,  they  may  both  b*  expunged,  and  the 
other  members  continually  multiplied  (as  by  the  rule) 
will  produce  the  fraction  required  in  lower  terms. 

6.  Reduce  §  of  |  of  j.  to  a  simple  fraction. 
Thus,  2x5 

1   ft     .     .     5  Q+mf 

—  i"i        T3T    **  ^^* 

4x7 
T.  Reduce  |*of  f  of  f  of  f£  to  a  simple  fraction. 


CASE  VI. 

To  reduce  fractions  of  different  denominations  to  equiva- 
lent fractions  having  a  common  denominator. 

RULE  1. 

1.  Reduce  all  fractions  to  simple  terms. 

2.  Multiply  each  numerator  into  all  the  denominators 
«xceptitsown,  fora  new  numerator;  and  all  the  denom- 
inators into  each  other  continually  for  a  common  denom- 
inator; this  written  under  the  several  new  numerators, 
vnll  give  the  fractions  required. 

EXAMPLES. 

I.  Reduce  |  «  |  to    equivalent  fra&tions,  having  a 
common  denominator. 

i  4-  •§  +  1=24  common  denominator. 


1 

2 

3 

X3 

2 

3 

3 

4 

9 

X4 

4 

2 

12 

16 

18 

new  numerators. 

24       24 
2.  Reduce  1 

24 
A  ai 

denominators.  ,' 
id  44  to  a  commc 

. 

5.  Reduce  £  f   *  and  £  to  a  common  denominator. 

S  tt  and 


100  REDUCTION  OF   VULGAR   FRACTIONS. 

4   Reduce  £  ^  and  -^  to  a  common  denominator. 

800      300          400 

-    -  and  -  =TV  ,V  and  ^  =  1-ft  Ana. 
1000     1000         1000 

5.  Reduce  >  £  and  124  to  a  common  denominator. 

/JWc     54    «  0     883 
^«S'  Ti    7"?     TI 

6.  Reduce  %  %  and  £  of  -fi  to  a  common  denominator.  ) 

/?MS        758      2592    1989 

tlln!>'  1STXS  7t?£  7TJS 


The  foregoing  is  a.general  Rule  far  reducing  fractions 
to  a  common  denominator;  but  as  it  will  save  much  la* 
iioar  to  keep  the  fractions  in  the  lowest  terms  possible, 
the  fallowing  Rule  is  much  preferable. 

RULE  II. 

For  reducing  function!  to  the  least  common  denominator? 

(By  Rule,  page  155)  find  the  least  common  multiple  of 
all  the  denominators  of  the  jiiven  fractions,  and  it  will  be 
the  common  denominator  required,  in  which  divide  each 
}>a;  iciriar  deuoininatov,  ui:d  multiply  the  quotient  by  its 
o\vn  numerator  for  a  new  numerator,  and  the  new  nume- 
rators being  placed  over  the  common  denominator,  will" 
the  fraction?  required  in  their  lowest  terms. 


EXAMPLES. 

1.  Reduce  1  1  and  f  to  their  least-  common  denemina* 
tor. 

4)2    4    8 


2)2     1     2 

1     1     1    4x2=8  the  least  c»m.   denominator 

8—2x1=4  the  1st.  numerator. 
8— 4xS=l  the  3d.  numerator. 
8—8x5=5  t'ie  3d.  numerator. 
These  numbers  placed  over  the  denominator,  give  the 
answer £ff  equal  in  value,  and  in  much  lower  terma 
than  the  general  Rule,  which  would  produce  f|  £!  TT 

2.  Rorluce  ^^  and  T7j  to  their  least  common  uenomi$ 
ii  a  tor. 


O7    VULGAK.    FRACTIONS.  l6l 


5.  Reduce  ^  f  f  and  Ty  to  their  least  common  de- 
nominator.         '  Am.  ||  T\  £|  || 

4.  Reduce  £  |  f  and  T'y  to  thtir  least  common  deiftm- 

||  if  TV 


ir-ator.  Ans. 


CASE  VII. 

To  reduce  the  fraction  of  one  denomination  to  the  fraction 
of  another,  retaining  the  same  value. 

RULE. 

Reduce  the  given  fraction  to  such  a  compound  one,  as 
will  express  the  value  of  the  given  fraction,  by  comparing 
it  with  all  the  denominations  between  it  and  that  denomi- 
nation you  would  reduce  it  to ;  lastly,  reduce  this  com- 
pound fraction  to  a  single  one,  by  Case  V. 

EXAMPLES. 

1.  Reduce  _£  of  a  penny  to  the  fraction  of  a  pound. 
By  comparing  it,  it  becomes  f  of  •£$  of  ^V  of  a  pound. 

5X1X1  5 

= Ans. 

6  x  12  x  20       1440 

2.  Reduce  -^5-  of  a  pound  to  the  fraction  of  a  penny. 
Compared  thus,  -j^Vtr  °f  2i°  °^  V*^- 

Then        5  x  20  x  12 

._  _  noo _.. j   JJfts 

440         1          1 

3.  Reduce  $  of  a  farthing  to  the  fraction  of  a  shilling. 

Ans.  jyS. 

4.  Reduce  f  of  a  shilling  to  the  fraction  of  a  pound. 

5.  Reduce  f  of  a  pwt;.  to  the  fraction  of  a  pound  tru^. 

6.  Reduce  |  of  a  pound  avoirdupois  to  the  fraction  of 
it  cwt.  Ana.  TiTc'«-i. 

7.  What  part  of  a  pound  avoirdupois  is  T-|?  of  a  cwt.  ? 
Compounded  thus,  ^e  of  4  of  ST«— !]••— f  Ans. 

8.  What  part  of  an  hour  is  j|T  of  a  week  r 

14* 


162  REDUCTION   O?  VULSAR  FRACTIONS. 

9.  Reduce  J  of  a  pint  to  the  fraction  of  a  hhd. 


W.  Reduce  -f  of  a  pound  to  the  fraction  of  a  guinea. 
Compounded  thus,  -$  of  3T°  of  ^.=4  Ans. 

11.  Express  5$  furlongs  ;n  the  fraction  of  a  mile. 

Thus,  5}=V  of  !=!£  Ms. 

12,  Reduce  |-  of  an  English  crown,  at  6s.  8d.  to  thfc 
fraction  of  a  guinea  at  28s.  »flns.  2\  of  a  guinea. 

CASE  VHI. 
i 

To  iind  tlie  value  of  a  fraction  in  tlie  known  parts  of  the 
integer,  as  of  coin,  weight,  measure,  occ. 

RULE. 

Multiply  the  numerator  by  the  parts  in  the  next  inferi- 
»u  denomination,  and  divide  the  product  by  the  denomina- 
K»r  ;  aad  if  any  thing  remains,  multiply  it  by  the  next  in- 
tv  rior  denomination,  and  divide  by  the  denominator  as 
before,  and  So  on  as  far  as  necessary,  and  the  quotient 
will  bo.  the  .1:1- 

Norii.  —  This  and  the  following  Case  are  the  same 
\ritli  Problems  11.  and  III.  pages  75  and  76  ;  but  for 
tlie  scholar's  exerci.-o,  I  shall  give  a  few  more  examples 

in  out1:. 

KXAMTLES. 

,'.  AVlvat  is  the  value  of  -|'-£  of  a  pound  ? 

3ns.  8s. 
2*  Find  the  valu*e  of  |  of  a  cvvt. 

^ns.  Sqrs.  3lb.  loz. 

3.  Find  tiie  value  of  |  of  5s.  6d-  .Ins.  3s.  Q$d. 

\.  Ilmv  mucli  is  TV^-  of  a  j)ound  avoirdupois  ? 

Jlns.  7  ox.  Wdr. 

nw  much  is  4  of  a  hhd.  of  wine  ?  .flns.  45^c/s. 
>\  \V  !iat  is  the  value  of  \\  of  a  dollar  ? 

AKS.  9$.  7±d. 
7.  What  is  tlie  value  of  f\.  of  a  guinea?   »0K* 


ADDITION   OF   VULGAR   FRACTIONS.  163 

9.  Required  the  value  of  ff£  of  a  pound  apothecaries. 

jins,  2oz.  Sgrs. 

9.  How  ranch  is  f.  of  5l.  9s.  ?      Jins.  £4  to,-.  5  |rf. 
10.  How  much  is  £  of  f  of  £  of  a  hogohead 

tins.  ISgalm.  3qts. 

CASE  IX. 

To  reduce  any  given  quantity  to  the  fraction  of  any  great- 

er denomination  of  the  same  kind. 
[See  the  Rule  in  Problem  III.  page  75.] 

EXAMPLES   FOR   EXERCISE. 

1.  Reduce  12lb.  Soz.  to  the  fraction  of  a  cwt. 

<?»«     is* 

"•I  Hi.   JTsTS 

£.  Reduce  13c\vt.  Sqrs.  2.0  Ib.  to  the  fraction  of  a  ton. 

.ins.  |» 

5.  Reduce  16s.  to  the  fraction  of  a  guinea.    3ns.  $ 
4.  Reduce  1  hhd.  49  gals,  of  wine  to  the  fraction  of  a 


- 
5.  What  part  of  4cwt.  Iqr.  24lb.  is  3c\vt.  Sqrs.  17lb. 


ADDITION  OF  VULGAR  FUACTIONS. 
RULE. 

REDUCE  compound  fractions  to  single  ones  ;  mixed 
numbers  to  improper  fractions  ;  and  all  of  them  to  their 
least  common  denominator  (by  Case  VI.  Rulo  II.)  then 
the  sum  of  the  numerators  written  over  the  coiim,->u  lie- 
nominator,  will  be  the  s,am  of  the  fractioi**  required. 

EXAMPLES. 


i.  Add  5i  5  and  f  of  {•  together. 

5i^=V  a 
Tlieu  y  ^  l±  reduced  to  their  least  LOU 

by  Gas*  VI.  Rule  II.  will  becmne  '- 
i  132-fl8+14=\fji=6^  or 


164  ADDITION   OF   VULGAR  TRACTIOKS. 

2.  Add  4  |-  and  £  together.  Ms.  1  £ 

S,  Add  1  1  and  £  together.  .ins.  !{• 

4.  Add  12$.  34  and  4|  together.  .5ns.  20}$ 

5.  Add  }  of  95  and  I  of  14$  together.  .fl/ts.  44^| 

NOTE  1.  —  In  adding  mixed  numbers  that  are  not  com- 
pounded vith  other  fractions,  you  may  first  find  the  sum 
of  the  fractions,  to  which  add  the  whole  numbers  of  the 
girea  mixed  numbers. 

•f.  Find  the  sum  of  JA  T*  and  15. 

I  find  the  sum  of  A  and  -*-  to  be  •£}•=• 

Then  1  »£+  o-f  r-HJ=-28.l£  Ans. 

r.   .\. 


"of  .;  of  tV  Jns.  3STJT 

NOTE  2.  —  To  add  fr.iciii'ns  of  money.  \\i':g];t,  &c.  re» 
lions  of  iliilerciit  integers  to  those  of  the  same. 

Or.  if  you  please  you  may  And  the  value  of  each  frac- 
VIII.  in  reduction,  and  then  add  them  in 
pn'jier  terms. 

9.  Acid  £  of  a  shilling  to  |-  of  a  pound. 


Method. 

*=r1< 
•  •  value  by  Case  V  III. 

,  Ud. 


2d  Method. 
|£.=73.  6d.  Oqrs. 
^s.  =0     (5     3| 

By  Case  VIII.  Reduction. 

10.  A«ia  I  Ib.  Troy,  to  f  of  a  mvt. 

•tins.  7o: 
» 1 .  Add  ^  of  a  ton,  to  T'ff.  of  a  cw  t. 

Ms.  IZcwt.  Iqr.  Sib.  IQ^oz. 
}  :1.  Add  |  of  a  mile  to  ^  of  a  furlong. 

ir.  £8»o. 
IS.  Add  ^  of  a  v;i:-;l.  ]  of  a  foot,  and  J  of  a  mire  to- 

.ins.  1540^c?s.  £//.  9in. 

.«!•!  '.  cif  a  work,  ^  of  aday,  $  of  an  hour,  and  $  of 
a  miiictc  together.  .jns.  2aa.  2/to.  SOmtn. 


SUBTRACTION    OF    VCLGA.R     FRACTION'S. 

SUBTRACTION  OF  VULGAR  FRACTION?. 

RU1 

PREPARE  the  frz  in  Addition,  and  the  dif- 

ftjrenceof  the  numerators  written  above  the  coir.wr.  de- 
nominator, mil  give  the  difference  of  the  tra.  ••  .  ed. 

EXAMi'LES. 

1.  From  $  take  4  of  £ 

I  of  1=^-^1  Tfc«"  -A  -rr 

Therefore  9— 7=«T2j  =  J  Me  jJns. 

2.  From  ^  take  *  J4 

3.  From  |j.  take  ft 

4.  From  14  take  -}}  13^7 

5.  What  is  the  difference  <  ."  ?  7*? 

6.  What  differs  -^  from  ^ 

7.  From  14$  take  f  of  19  lT7j 

8.  From  ££  take  0  remains. 

9.  From-j-i-  of  apcund,  take  -J  of  a  shilling. 
fofiV^ifc^'^enfrom^.tal 

No'iE. — In  tractions  oi  mo:iey,  weight,  &.c.  you  inav,  i^ 
xtou  pleasejiind  thevalu  .veu  iractidiis  (by  Gis^ 

Vlll.  in  Reduction)  and  then  subtract  them  iu  their  pro- 
per terms. 

10.  From  T7T£.  take  3|  shilling.  Jns.  5s.  M.  ^}qrs. 

11.  From  |  ot*  an  oz.  take  |  of  a  ; 

.   llpit't.  ?^?%. 
1£.  Frora  A  of  a  cwt.  take  -^  of  ;i  II). 

s.  for.  QTlb.  Coz.  Wfclr. 

13.  From  3|  \veeks,  take  I  ot  a  day,  and  i  of  f  of  ^  of 
an  hour.  .i»s.  Gtc.  -Ida.  12/io.  19.'/it/j.  ir.}sec. 

*In  3ubtracting  mixed  numbers,  when  the  lower  fraction  ia 
greater  than  !l:e  upper  one,  you  may,  without  reducing  them 
to  improper  fractions,  subtract  the  .numerator  of  the  lower 
fraction  frojn  the  common  denominator,  and  to  that  <]i[iereuce 
add  the  upper  numerator,  carrying  one  to  the  suit's  place  of 
the  lower  whole  number. 

Also,  a  fraction  may  be  subtracted  from  a  whole  number 
by  taking  the  numerator  of  the  fraction  from  ina- 

tor,  and  placing  the  remainder  over  the  denominator,  then 
taking  one  from  the  whole  number. 


166  MULTIPLICATION,    DIVISION,   StC. 

MULTIPLICATION  OF  VULGAR  FRACTIONS. 
RULE. 

REDUCE  whole  and  mixed  numbers  to  the  improper 
fractions,   mixed  fractions  to  simple,  ones,  and  those  ot 
lifferent  integers  to  the  same  ;  then  multiply  all  the  nu- 
merators i  for  a  new  numerator,  and  all  the  de- 
nominators together  for  anew  denominator. 

EXAMPLES. 

1.  Multiply  $  by  -*-  Answers.  }|=4 

2.  Multiply  $  by  2-  •}£ 
.  Multiply  5i  by.  $ 

4.  Multiply  5.  of"7  by  *  3}$ 

:•.  Multiply  j^.?-  by  ->5 

G.  Muli'              :  S  by"]  of  5  is| 

7.  Multiply  7j  by  9*  69| 

8.  Multiply  I  of  %  by  g.  of  S?  |^ 

9.  What  is  the  continued  product  of  £  of  ^,  7,  5 4  and 

/> 


DIVISION  OF  VULGAR  FRACTIONS. 
RULE. 

PREPARE  the  fractions  as  before;  then,  invert  the 
ufvisor  and  proceed  exactly  a*  in  multiplication: — TTfe 
products  will  be  the  quotient  required. 

EXAMPLES. 

4X5 

1.  Divide  \  by  j  Thus, =|?  An$. 

:>  x  r 

2.  Divide  ',  7r  In-  ?                              Jlnsicers.  l-2^ 

3.  Divide  -jj  of>  I  |v 
1.  ^"l>at  is  the  quotient  of  17  by  I  ?  59| 

7| 
ivide  *  of  »  of  i  by  1  of  } 

7.  Divide  4.?,-  by'  f  of  4 

8.  Divide  71  by  127  T7fV 

9.  Divide  5*o4  b^  I  of  51 


BTJI.E    O7    THUEB    DIRKCT,    INVKRSfc.    &C. 

RULE  OF  THREE  DIRECT  IN  VULGAR 
FRACTIONS. 

RULE. 

PREPARE  the  fractions  as  before,  then  state  your 
question  agreeable:  to  the  Rules  already  laid  down  in  the 
Rule  of  'i'h  roe  in  \vlxm  .  ,  and  invert  the  fi'-st  term- 

in  t'.ie  proportion  ;  tlien  multiply  all  the  three  terms  con- 
tinually together,  and  the  product  will  be  the  answer,  iu 
the  sau.e  name  with  the  second  or  middle  term. 


1.  If  |  of  a  yard  cost  2-  of  a  pound,  what  will  ?s  of  an 
Ell  English  cost  .- 

|vd.=Jof  f  <»f  £=|g  or}  Ell  Eiiirlis'i. 
Ell.  £.  '  Ell.  s.'d.  qrs. 

::,'V    Wi^fXA-M^'-W  «    M 

i.  11  -;  c,i  a  van!  cost  ^  ot  a  pound,  what  will  4<< 
etune  '  jfns.  £59  Ss.  (i]^. 

3.  If  50  bushels  of  wheat  cost  17£/.  whai  is  it  ]>er  bas!»- 


4.  If  a  pistareea  be  wortii  1-1^-  ^.ence,  whut  are  1 
tarcens  worth  ?  «tf;j  .-. 

5.  A  merchant  sold  JA  j-iecc-sof  cloth.  •  ••.minj 
iM^.  vd.->.  at  (Js.  Id.  [>er  yard  :  v/hat  did  the  whole  ;, 

to?  Jus.  /JGO   10>-.  08 

C.  A  person  having  •;  o'f  n  \es-ei.'relU  |  of  i 

.  :  what  is  iho  \\  '  i  worth  ?  °     ..*»/.«?.  / 

7.  II'7  (tf.as!'.;p  bo   worilj   ft  of  her  cargo,  valued   a* 
\k!uit  is  Uie  whole  ship  and  car^o  worth? 
.  /MOOSl   14b'.   i! 


SE  PROPORTION. 
IliM.K. 

PREl'ARE  the  fractions  a:i'l  ^.«<  IK-- 

fore,  then  invert  the  third  term,  ;I;H! 
terms  together,  tl.e  product  will  b«  •.,  t:-. 


J68  RULE   OF  THn"EE   DIRECT  Itf  DECIMALS 

EXAMPLES. 

1.  l!o\v  much  si  i  a)  loon  that  is  |yard  wide,  will  line  5£ 

ol  cloth  which  is  1  }  yard  wide  r 
Yds.  7/i's.  yds.  Yds. 

As    1|  :  ;xVxf  =  V45=1G:jV^"s- 

2.  If  n   ;  i^n  perform  n  jc-nm-y  in  S|  days,  when  the 
<jay  i  :  i  how  many  days  mil  he  do  it 

iiirs.  j9«y.  4  gVj  rfflj/s. 

;•-.  s,  mow  21  Jatres,  in  now  many 

.fltts.  18||  </«^.s\ 
4.  <at  is  7-}  inches  broad,  will 

Jlns.  £0  inches. 

.>.  ;•  carriage  of  an  c\vt.   145$ 

ovt.  be  carried  for  the  same  mo- 
ney r  Jins.  22^  miles. 

6.  How  pi.-xny  yaids  of  baize  which  is  U  yards  v»ide, 
will  line  18£  yanls  of  camblet  g-yd.  wide? 

is.  1 1^/i/s.  Igr.  1}  no. 


RULE  OF  THREE  DIRECT  IN  DECIMALS 

HULL;. 

IsEDl.TE  your  '  .decimals,  and  state  your 

ijuo'io:.  as  in  uin;>ly  the  second   and 

third  together ;  divide  by  the  iirst,  and  the  quotient  will 
be  the  answer,  £c. 

vMl'LES. 

1.  If  I  of  a  yd.  cost  •/?  of  n  pound  ;  what  will  15$  yds. 
come  to?  |  =,875  yr=,5«3-fand  3=,75 

J''s-      L-  £•  s-  d-  Hrs- 

As  ,K75    :    ,583    :  :    13,75  :  10,494  =  10  9  10  2,24  Ans 

2.  If  I  pint  of  wine  cost  1,2s.  what  cost  12,5  hhd«.  ? 

Ans.  £.378 
5.  If  4*ydi,  cost  3».  4>d.  >vhat  will  30J  yds.  cost  ? 


SIMPLE    INTEREST   BY    DECIMALS.  169 

4.  If  1.4cwt.  of  sugar  cost  lOdols.  9  cts.  what  will  9 
cut.  3  qrs.  rost at  the  same  rate  ? 

eu-t.         S  <*K-f.     8 

As  1,4  :  t  10,09  :  :  9,75  :  rO,S69«=g:0,  26c£s.  9m. -j-; 

5.  It'  19  yards  cost  25,75  dols.  what  will  435$  yards 
come  to  r  .3ns.  8590,  Slcte.  7,a5m. 

G.  If  545  yards  of  tape  cost  5  dols.  17  cents.  5m.  what 
will  1  yard  "cost?  .fins.  ,015=lJcJs. 

7.  if  a  man  lay  s  out  121  dols.  £5  cts.  in  merchandize, 
and  thereby  gains  39,51  dols.  how  much  will  he  gam  in 
laving  out  1£  dollars  ui  the  >ame  rate  ? 

Jns.  ?,<>i  d»k.*=g3,  91 

c.  Howiuany  yards  of  ribbon  can  1  buy  for  25; 
if  x9^  yds.  cost  4}  dolla  Jn-f.  178J?/ajv/s. 

9.  It  178i  yd:-.c<»^  £5]  dollars,  what  cost  £9$yaids? 

Jus.  g4i 

10.  If  1,6  cwt.  of  sugar  cost  12  dols.  12  cts.  what  cost 
S  Uhds.  each  11  cwt.  3  qrs.  10,1 '2  Ib.  ? 

J.-iS.  269,07 


M  M  P  L  E  1 N  T  E  R  E  S  T  Ti  Y  1)  E  C I M  A  L  ,S. 
A  TABLE  OF  RATIOS. 


Hate  per  cent. 

f  io.      j  jKa^e  per  cent.  \      Re 

•  5 
4 
4J 

5 

,"03 

,04 
,045 
,05 

6 

$1 

. 

,06 
'      ,065 
,07 

Ratio  is  the  simple  interest  of  1^.  for  ono  year  :  or  in 
federal  money,  of  gl  £»;•  OUR  year,  at  the  rate  per  cent, 
agreed  on. 

RULE. 

Multiply  the  Principal.  Ratio  and  time  con 
gether,  aud  the  last  product  will  be  the  interest  rc(;u.. 

EXAMPLES. 

1.  liccjuired  the  interest  of  £1 1  doh.  4i  cts.  far  5 ; 
at  5  per  cent,  per  annum  ? 


'SIMPLE   INTEREST   BY    DECIMALS. 

3  cL;. 
211,45  Principal. 

,05  Ratio. 


10,5725  Interest  for  one  year, 
o  Multiply  by  the  time. 


52,86'2.i  Jns.=g52,  86c?s.  2 
2.  \Vhat  is  the  interest  of  6451. 10s.  for  3  years,  at  6 
per  cent,  per  annum  ? 

xO«xf>=l  IO,190=£11G  Ss.  9<7.  2,4grs.  .Jns. 
What  is  the  interest  of  \-2lL  8s.  6d.  for  4i  years, at 
6  per  cent,  per  annum?  l.GGnrs. 

4.  \Vh;it  is  tliii  amount  of  506  dollars  39  rents,  for  1$ 
years,  at  G  per  cent..  per  annum  ?  -.  8584.6651 

5.  Required  the  amount  of  (>43  tloU.  50  cts.  for  12jyrs. 
at  5 A  nor  c^ut.  per  annum  ?          Jns.  gl!03,  26c/s.-f 

i;  ii. 

Tlie  amount,  time  and  ratio  given,  to  find  the  principal. 

RULE. 

Multiply  the.  ratio  by  the  time,  add  unity  to  the  product 
fur  a  divisor,  by  which  sum  divide  the  amount,  and  th« 
quotient  \\ill  be  the  principal. 

F.KAMPI.F.S. 

1.  What  principal  will  amount  to  1235,975  dollars,in 
5  years,  at  6  per  cent,  per  annum  ?     g  55 

'  ,06x5  + 1  =  1 ,30)  1235,975(950,75  Jtwt. 

2.  \Vhatprincipal  will  amount  to  87o/.  19s.  in  9 years, 
at  fi  per  cont.  per  annum  r  «*/ns.  £567  10s. 

...  \\'l<!,t  principal  will  amount  .^Is.  6  cts.  in  13 

years,  at  7  per  cent.  ?        Jlns.  Y  ,  25r/s. 

4.  AVIiat  principal  will  amount  to  fof;.'.  10s.  4.1S5d. 
in  8|  years,  at  5-i  por  cent.  ?  J)ns.  £645  15s. 

K  III. 
The  amount,  principal  and  time,  given,  to  find  the  ratio, 

h'l'hK. — Siutract  the  pj-incipul  from  .he  amount,  d-i* 
>i:le  thevema:  ••  pi-odiict ;  of  the  tiine  and 

pal,  and  the  quotient  will  be  the  ratio. 

KXAMPLK3. 

1.  At  what  rate  per  c;rnt.  will  950>75  doll,  amount 
1235,975  dols»  in  5  yours? 


SIMPLE    INTEREST   BY    DEC1MAM-  i7\ 

From  the  amount    =    1235,97.; 
Take  the  principal  =      950,75 

950,75  X5=4753,75)£85,225()(,06=tj  per  cent 
285,2250  J/iS. 

2.  At  what  rate  per  cent,  will  j'GTY.  10s.  amount  to 
87S/.  19s.  iu  9  years  ?  Jtns.  6 ptr  ant. 

3.  At  what  rate  per  cent,  will  340  dols.  25  cts.  amount 
to  626  dols.  6  cts.  in  r"  voars?  vins.  7  per  cent. 

4.  At  what  rate  per  cent,  will  G45/.  15s.  amount  to 
<J56/.  10s.  4,125d.  in"  84*  years  ?          .flns.  5  J  per  cent. 

^E  [V. 

The  amount,  principal,  and  rate  per  cent,  given,  to  find 

the  time. 

RULE. 

Subtract  the  principal  from  the  amount;  divide  the 
remainder  by  the  product  of  the  ratio  and  principal  j  and 
the  quotient  will  bo  •: 

KXAMl'f.1,5. 

1.  In  what  time  will  9;'Udols.  75  cts.  amount  to  13. ".3 
dollars,  97,5  cents,  at  G  per  cent,  per  annum  ? 
From  the  amount     ^1235.975 
Take  the  principal       950,75 

950,75x06=57.0-150)285,2250(5  years,  Ms, 
,  .i250 


2.  In  what  time  will  5C7/.  10s.  amount  to  873/.  19s. 
at  6  per  cent,  per  annum  t  Jlru.  9  years. 

5.  In  what  ti'..ie  will  340  dols.  25  cts.  amount  to  6i6 
dols.  6  cents  at  7  per  cent,  per  annum  r  *ins.  12  years, 

4.  In  what  time  will  G45/.  15.-..  amount  to  956/.  10$. 
4,125d.  at  5}  per  ct.  per  annum  r  vi/js.8.75=b^  years. 

TO  CALCULATE  INTEREST  FOR  DAYS. 
&JLE. 

Multiply  the  principal  by  the  given  number  of  days, 
^nd  that  product  by  the  ratio;  divide  the  last  product  by 
36.5  (the  number  of  days  In  a  year)  and  ii  will  give  th* 
interest  required 


SIMPLK   INTCRKST  BY  DECIMALS. 


EXAMPLES. 

1.  What  is  tlie  interest  of  SGOf.  10s.  for  146  days,  at* 
per  cent.  ? 

S60,5xl46x,06      £.      £.s.  rf.  grs. 

=8852=8  13  0  1,9  Ans. 

365 

•2.  "What  is  the  interest  of  G40  doJs.  60  cts.  for  100  days 
at  6  percent,  per  annum  ?  Jns.  g!0,53c's.-f- 

3.  Required  the  interest  of  250/.  17s.  for  120  ilavs  at 
5  per  cent,  per  annom  ?      Jns.  £4,1235=4.'.  25.  5\d.+ 

4.  Required  the  interest  of  4Sf  dollars  75  cents,  for  25 
dajs,  at  7  per  cent,  per  annum  r     »tf,r--.  8 -,  SOefs.  9w.-f 


?  ^ 

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1 

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§  §J 

it 

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co     c 

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SIMPLE    INTEREST   BY    DECIMALS. 


When  interest  is  to  be  calculated  on  cash  accounts,  &c. 
where  partial  payments  are  made  5  multiply  the  several 
balances  into  the  days  they  are  at  interest,  then  multiply 
the  sum  of  these  products  bv  the  rate  on  the  dollar,  ami 
divide  the  last  product  by  365,  and  you  will  have  the 
whole  interest  d-.e  on  the  account,  otc. 

.  \  MPLES. 

Lent  Peter  Trusty,  per  bill  on  demand,  dated  1st  of 
June,  1800,  2uOO  dollars,  of  which  I  received  back  the 
19th  of  August.  400  dollars  ;  on  the  15th  of  October,  CiK- 
dollars:  on  the  llth  of  ijecember,  400  dollars  :  on  the 
17th  of  February,  1801,  200  dollars;  and  on  the  1st  of 
June,  400  do  .'-lUch  interest  is  due  on  the  bill, 

reckoning  at  G  ..-cr  ci  i.i.  ? 

1  800,  dulls,  days,  products, 


June  1,  Principal  per  bill, 

August  19,  Received  in'  part, 

Balance. 
October   15,  Received  in  purt, 

Balance, 

December  11,  Rcceued  iii  part, 

1801,                          l:.tlanoe, 
February  17,  Received  in  part. 

Balance, 
Jiir>e  1,  Ree'd  in  full  of  prim  : 

Then  333600 
,U6  Ratio. 

2000 
400 

79 

57 
57 

104 
«  i 

15800U 
91200 
57000 
40^00 
41  GOO 

1C-  0 

iOOO 
400 

600 

200 

400 
400 

| 

S88600 

• 

3U.5V2331 6,00(63,879  .IMS.  =  63  87  9 -f- 
iH'iirh/r  llrh  f<ir  comp^ti)i<r  ivtcrett  !.»•  nr»/  nnt«, 
or  obligation*  n-ti<>;i  tiu*re  are  payment*  ui  purl.  i>r  >-iul'tp*se- 
mei;  •'•/  Lite  tiiijwiur  Co;  ''itai-e, 

'of  Cunneciict;!,  in  1  78  \. 

RULE. 

"  Compute  ti.ji  i.iierest  to  the  time  of  the  iir-t  ::ftv- 
' 


174  'SIMPLE  IXTEREST  BY  DEOIMA. 

mcnt ;  if  that  he  one.  vear  or  more  from  the  time  tiie  in- 
terest commenced',  add  it  to  the  principal,  an.'  deduct  the 
payment  from  the  sum  total.  If  there  be  afterpayments 
made,  compute  the  interest  on  the  balance  due  t •• 
next  payment:,  and  then  deduct  the  payment  as  above; 
and  in  like  manner  fnnn  one  payment  to  another,  till  all 
the  payments  are  absorbed;  provided  the  time  between 
one  payment  and  another  be  one  year  or  more.  But  if 
any  payment  be  made  before  one  year's  in;  i  ac 

crucd,  then  compute  the  interest  on  the  ; 
due  on  the  obligation  for  one  year,  add  it  to  ihe  principal, 
and  compute  t!iO  interest  on  the  sum  paid,  from  the  time. 
it  was  paid,  up  10  the  end  of  the  year;  add  it  to'the  sum 
pau!,  and  deduct  that  sum  from  the  principal  and  interest 
added  as  above.* 

••  If  any  payments  he  made  of  a  lass  sum  than  the  in- 
terest arisen  at  tiie  time  *.('  such. payment,  no  interest  U 
to  be  computed  but  only  <,u  the  principal  sum  fur  a^r 
period."  K'irht/'s  li  p:>rt*. /v/^r 

AMi-i.r.s.  *• 

A  bond,  or  :  d  January  -4t!i,  1707,  was  j^ivcn 

for- 1600  dollars,  Mite  rest  at  i  per  cent,  and  there  were 
payments emlorsed  upon  it  a.->  follows,  >  8 

1st  payment  }V:,mai  v  1J,  i  ^UO 

2d  pa/meht  June  ^','  ir'Ji>.  500 

3d  payment  November  14,  1799.  260 

I  demand  how  111110:1  remains  due  on  said  iu»tG  the  24tl) 
of  December,  J 

1000,00  dated  January  -1,  1797. 
67,  19,  1798ir=lSi  months^ 

1067,50  amount.  [Carried  tt| 

*lf;i  y c.-'r  (lues  IMP!  cxtei  ii   l-i-yond  l^c  t:;.  <  ••'tle- 

mri.l  :  ,.    ;i,iiuunl  i,.  U. 

due  on  ;!if  ol)li^,  .me  of  s«.-Mlcm« ; 

I'lMtJ  the  aini<itiit  of  itic  ^inji  p.iid,  from  the  time 

to  the  time  n!"  i'mal  neHlejiirut,  and  deduct  tliij  an:- 

the  amo:ii:t  c.f  the  jii-::.ci,.a!.     l'>Ht  iftliere  be  Hevi-ra!  payments 

made  \\ithiutlie  .  find  the  amount  of  the  several  \ 

iwi.l  ,  tVom  the  time  t!i«-y  \\crt-  paid,  to  tiie  tiine  ol'selUeiiifiit, 

and  deduct  their  amount  .mount  of  ll.e  , 


SIMPLE   INTEREST  BY   DECIMALS.  175 

I06f  ,50    amount.  [Brought  Up. 

200,00    first  payment  deducted. 

3jti7,50    balance  due,  February  19,  1798. 
70,845  Interest  to  June  29,  1799 «s 16$  month*. 

958,345  amount. 

§00,000  second  payment  deducted. 


458,545  balance  due,  June  29, 
£6,30     Interest  for  one  year. 

464,645  amount  for  one  year. 

£69,750  amount  of  third  payment  for  7\  months.* 

194,895  balance  due  June  £9,  1800.       mo.  Aa. 
5,687  Interest  to  December  £4,  1800,  5    25 


200,579  balance  due  on  the  Note,  Dec.  24,  1800. 
RULE  II. 

Established  by  the  Courts  of  Law  in  Massachusetts  for 
computing  interest  on  notes,  <§"c.  on  which  partialpay- 
mtnts  have  been  endorsed. 

"  Compute  the  interest  on  the  principal  sum,  from  the 
time  when  the  interest  commenced  to  the  first  time  when 
a  payment  was  made,  which  exceeds  either  alone  or  ia 
conjunction  with  the  preceding  payment  (if  any)  the  in- 
terest at  that  time  due  :  add  that  interest  to  the  princi-* 
pal,  and  from  the  sum  subtract  the  payment  made  at  that 
time,  together  with  the  preceding  payment  (if  any)  aad 
the  remainder  forms  a  new  principal ;  on  which  compute 
and  subtract  the  payments  as  upon  the  first  principal, 
and  pmceed  in  this  manner  to  the  time  of  final  settle- 
ment." 

S  cts. 

*260,00  third  payment  with  its  interest  from  the  time  it 

9,75        was  paid,  up  to  the  end  of  the  year,  or  front 

./You.  14,  1799,  to  June  29,  ?800,  which  is  7§ 


176 


SIMPLE    INTEREST    BY    DECIMALS. 


Let  the  foregoing  example  be  solved  by  this  Rule. 

A  note  for  1000  dols.  dateu  Jan.  4,  179",  at  6  per  ceut 
1st  payment  February  19,  1793.  g200 

2U  payment  June  29,  1799.  500 

3d  payment  November  14,  1799.  260 

How  much  remains  due  oil  said  note  the  24th  of  Do- 

:cmber,  1800?  8  cl*. 

Principal,  January  4,  1797,  1000,00 

Interest  to  Feb.  19, 1798,  (IS*  wo.)  07,50 


Paid  February  19,  1798, 


Amount,     lfifS7,.v> 
200,00 


Remainder  for  a  new  principal,  867,50 

Interest  toJune  £9,  1799,  (lfi£  me.)  70^84 


Paid  June  29,  1799, 


Amount,    938,54 
500,00 


Remains  for  a  nt 

Interest  to  November  J4,  1799,  (4}  mo.) 


438,34 
9,86 


November  14,  If96,  paid 
Remains  a  new  principal 


Amount,    448,20 
260,00 

'  188,20 

Interest  to  December  24,  1800,  (IS*  mo  )        12,70 

Balance  due  on  said  note, Dec.  24,  1800,        £00,%< 

g  rN. 

The  balance  by  Rul«  I.    200,579 
Byllule  IJ.    200,990 

Difference,     0,41 1 

Anotler  Example  in  Rule  H. 

A  l>*>nd  or  note,  dated  February  1,  ItiOO,  \va«^i>*a  h" 
J(»»>  dollars,  interest  at  G  per  cent,  and  ihc,  • 

upon  it  as  follows,  vi/,.  fc  Cl 

l>;  j.aviueut  May  1,  1800, 
2d  (  "  •urcmber  14, 1 


COMPOUND    INTEREST    15Y    DF.C'IMALS.  177 

3d  payment  April  1,  1801.  12,00 

4th  payment  May  1,  1801.  50,00 

How  much  remains  due  on  said  note  the  16th  of  Sep. 

tomber,  1801?  S  ets. 

Principal  dated  February  1,  1800,  500,00 

interest  to  May  1,  1800,  (5  mo.}  7,50 

Amount,    507,50 
Paid  May  1,  1800,  a  sum  exceeding  the  intu  -~i,t.  40,00 

New  principal,  May  1,  1800,  467,50 

futerest  to  May  1,  1801,  (1  yenr.)  28,05 

Amount,    495,55 
Paid  N«.v.  4,  1800,    a  sum  less  than  the 

interest  then  due,  8,00 

Paid  April  1,  1301,       do.        do.      12,00 
Paid  May  1,  1801,  a  sum  greater,      30,00 

50,00 

New  principal  May  1,  1801,  445,55 

Interest  to  Sep.  16,  1801,  (4}  mo.)  10,02 

Balance  due  on  the  note,  Sept.  16,  1801,     $455,57 
|Cp  The  payments  being  applied  ace  or  ding  to  this  Rult} 
keep  down  the  interest,  and  no  part  of  the  interest  ever 
forms  a  part  of  the  principal  carrying  interest. 

COMPOUND  INTEREST  BY  BECIMALS. 
RULE. 

MULTIPLY  the  given  principal  continually  by  the 
amount  of  one  pound,  or  one  dollar,  for  one  year,  at  th« 
rate  per  cent,  given,  until  the  number  of  multiplication* 
are  equal  to  the  given  number  of  years,  and  the  product 
will  be  the  amount  required. 

OK,  In  Table  I.  Appendix,  find  the  amount  of  one  dol- 
lar, or  otie  pound,  for  the  giren  number  of  years,  which 
multiply  by  the  given  principal,  and  it  will  give  the 
amount  as  before. 


1*8 


EXAMPLES. 


1  .  What  will  400Z.  amount  to  in  4  years,  at  6  pet  cent. 
per  annujn,  compound  interest  ? 

400xl,06xl,06xl,06xi,06=:£504,99-f  or 

[£504  19s.  9d.  2,75jrs.+  Ms. 
Tlie  same  by  Table  I. 
Tabular  amount  of  £1  =  1,26247 
Mullipiy  by  the  principal  400 

Whole  amount—  £5t)4,98800 

2.  Required  the  amount  of  4i.5  dol».  75  cts.  for  3  yearf, 
at  6  per  cent,  compound  interest.        »3?w.  $507,7ic/s.-f 

3.  Vv  hat  is  the  cump  -iT.nl  interest  of  555  dols.  for  14 
years,  at  5  percent.,  r     By  Table  I.     dns.  $$548,86c£s.-}- 

4.  What  will  50  dollars  amount  to  in  20  years,  at  6  per 
cent,  compound  interest  .'          dns.  fc>!60  Sects.  G^m. 


INVOLUTION. 

IS  the  ini'ltiplyin^;  any  number  with  itself,  and  that  pro- 
duct ty  tK*;  farmer  multiplier  ;  and  so  ou  j  and  the  several 
pro'luct*  which  arise  are  called  powers. 

The  number  denoting  the  height  of  the  power,  is  called 
the  index,  or  exponent  of  that  power. 

EXAMPLES 

What  is  the  5th  power  of  8  ? 
8  the  root  or  1st  power. 


64  =at  £d  power,  or  square 
8 

512  =a  3d  power,  qr  cube. 
8 

4096  =  4th  power,  orbiquadrate. 
8 

CrCS  s«  5th  power,  or  siu-salid.    A**. 


179 


What  is  the  square  of  17,1  ?  Ms.  292,41 

What  is  the  square  of  ,085  ?  Ans.  ,007225 

What  is  the  cube  of  25,4  ?  Ans.  16387,064 

What  is  the  biquadrate  of  12  ?  Ana.  20736 

What  is  the  square  ot'Ti  ?  .5ns.  52^ 


EVOLUTION,  OR  EXTRACTION  OF  ROOTS. 

WHEN  the  root  of  any  power  is  required,  the  busi- 
ness of  finding  it  is  called  the  Extraction  of  the  Root. 

The  root,  is  that  number, 'which  by  a  continual  multipli- 
cation into  itself,  produces  trie  given  p'>wer. 

Although  there  is  no  number  but  what  will  produce  a 
perfect  power  by  involution,  yet  there  are  many  numbers 
of  which  precise  roots  can  never  be  determined.  But,  by 
the  help  of  decimals,  we  can  approximate  towards  the 
root  to  any  assigned  degre^  of  exactness. 

The  roots  which  approximate,  are  called  surd  root?, 
and  those  which  are  perfectly  accui  ate  are  called  rational 
roots. 

A  Table  of  the  (Squares  mid  Cubes  of  the  nine  digits. 


Roots. 

\  1 

1  2 

3 

u 

5 

61 

7 

8 

9 

Squares. 

M 

1  4 

9 

16  J 

36  | 

49 

64 

81 

Cubes. 

M 

l« 

27 

64  | 

125 

216  | 

343 

512 

729 

EXTRACTION  OF  THE  SQUAPE  ROOT. 

Any  number  multiplied  into  itself  produces  a  square. 

To  extract  the  square  root,  is  only  to  nnd  a  number, 
which  being  multiplied  into  itself,  shall  produce  tne  given 
number. 

RULE. 

1.  Distinguish  the  given  number  into  periods  of  two 
figures  each,  by  putting  a  point  over  the  plan  of  units, 
another  over  ti1*  r/.aoe  >ds,  an-1  -  ui  if 
there  are  dyirnals,  point  th^rv  in  the  same  manner,  from 
units  towards  the        M  hand;  which   points  bhow  tha 
number  of  figures  the  root  will  consist  of. 

2,  Find  the  greatest  *MU  are  number  in  the  fiist,  orient 


180 

liand  period,  place  the  root  ot  it  at  the  right  hand  of  the 
given  number,  (after  the  manner  of  a  quotient  in  division) 
tor  the  first  figure  of  the  root,  and  the  square  number 
under  the  period^  and  subtract  it  therefrom,  and  to  the 
remainder  bring  down  the  next  period  for  a  dividend. 

3.  Place  the  double  of  the  root,  already_found,  on  the 
left  hand  of  the  dividend  fora  divisor. 

4.  Place  s«ch  a  figure  at  the  right  hand  of  the  divisor, 
and  also  the  same  figure  in  the  root,  as  when  multiplied 
into  the  whole  (increased  divisor)   the  product  shall  be 
equal  to,  or  the  next  less  than  the  dividend,  and  it  will  be 
the  second  figure  in  the  root. 

5.  Subtract  the  product  from  the  dividend,  and  to  the 
remainder  join  the  nexJ,  period  for  a  new  dividend. 

G.  Double  the  figures  already  found  in  the  root,  for  a 
ilexv  divisor,  and  from  these  find  the  next  figure  in  the 
root  as  last  directed,  and  continue  the  operation  in  the 
same  manner,  til!  you  h  I  own  nil  the  periods. 

Oi,to  facilitate  the  foregoing  Rule,  wla-M  you  have 
brought  d i. wa  a  period,  and  forme  I  a  dividend,  in  order 
to  find  a  new  figure  in  the  root,  you  may  divide  said  divi- 
dend, (omitting  the  right  hand  iigure  thereof,)  by  double 
the  root  already  founn,  and  the  quotient  will  commo  !y 
be  the  figures  sought,  o,-  heing  ::ntU:  le«s  o:ie.  or  two,  will 
generally  give  the  next  figure  in  the  quotient. 


EXAMPLES. 


1.  Required  the  square  root  of  t -11 225,64. 

141225,64(375,8  the  root  exactly  without  a  remainder ; 

9  but  when  the  periods  belonging  to  anv 

—  given  number  are  exhausted,  and  still 

67,512  leave  a  remainder,  the  operation  may 

469  be  continued  at  pleasure,  bj 

periods  oi  cyphers,  £c. 

745)4325 
5725 


7508)60064 
60064 


0  remain* 


EVOLUTION,  Oft   EXTRACTION   OF   ROOTS.         181 

Answers. 

2.  What  is  the  square  root  of  1296  ?  36 

3.  Of        56644  ?  £3*8 

4.  Of   5499025  ?  £345 

5.  Of   36372961  ?  6031 

6.  Of   184,2?  13,57-f 

7.  Of        9712,695809  ?  98,553 

8.  Of        0,45369?  j673-f 

9.  Of        ,002916?  ,054 

10.  Of        45  ?"  6,708-ji 


TO  EXTRACT  THE  SQUARE  ROOT  OF 

VULGAR  FRACTIONS. 

RULE. 

Reduce  the  fraction  to  its  lowest  terms  for  this  and  all 
other  roots ;  then 

1.  Extract  the  root  of  the  numerator  for  a  new  nume- 
rator, and  the  root  of  the  denominator,  fora  new  denomi- 
nator. 

2.  If  the  fraction  be  a  surd,  reduce  it  to  a  decimal,  and 
extract  its  root. 

EXAMPLES. 

1.  What  is  the  square  root  of  -^  ?  Jlns.  £ 

2.  What  is  the  square  root  of  /^  ?          Jlns.  %% 

3.  What  is  the  square  root  of  -*4|  ?  .   *flns.  £ 

4.  What  is  the  square  root  of  20|  ?  Jltis.  4| 

5.  What  is  the  square  root  of  248  TV  ?      Jns.  ISf 

SURDS. 

6.  What  is  the  square  root  of  ££?          Jlns.  9128-f- 

7.  What  is  the  square  root  of  ^|  ?         Jlns.  ,7745  4- 

8.  Required  the  square  root  of  361  ?      Jhs.  6,0207+ 


APPLICATTON  AND  USE  OF  THE  SQUARE 
ROOT. 

PnoitLF'-f  T.     A  certain  General  nets  nnai-myot  5184 
men;  how  many  must  he  place  in  rank  ant  $e,  to  form 

•'""Ti    -nto  a  square? 

1  P 


183          EVOLUTION,   OH   EXTRACTION    OF    ROOTS. 

RULE. 

Extract  the  square  root  of  the  given  number. 

v/5184="2  Jim. 

PHOB.  II.  A  certain  square  pavement  contains  20736 
square  stones,  all  of  the  same  size  ;  I  demand  how  many 
are  contained  in  one  of  its  sides  ?  ^20756=144  Ans. 

PROB.  III.  To  find  a  mean  proportional  between  two 
numbers. 

RULE. 

Multiply  the  given  numbers  together,  and  extract  the 
square  root  of  the  product. 

KXAMPLF.?. 

What  is  the  mean  proportional  between  18  and  72  ? 
72x18=1290,  and  v/ 1296 =36  Ms. 

PROB.  IV.  To  form  any  body  of  soldiers  so  that  they 
may  be  double,  triple,  &c.  as  manv  in  rank  as  in  file. 
HULK.    ' 

Extract  the  square  root  of  1-2,  1-3,  &c.  of  the  given 
number  of  men,  and  that  will  be  the  number  of  men  in 
file,  which  double,  triple,  &c.  and  the  product  will  be  the 
number  in  rank. 

EXAMPLES. 

Let  13122  men  be  so  formed,  as  that  the  number  in 
rank  may  be  double  the  number  in  file. 

13122-5-2=6561,  antl  v/GoGl^Sl  in  file,  and  81x2 
=  162  in  rank. 

PROB.  V.  Admit  10  hlids.  of  water  arc  discharged 
through  a  lead  en  pipe  of  2.}  inches  in  diameter,  in  a  cer- 
tain tnr.e;  I  demand  what  the  diameter  of  another  pipe 
must  be,  to  discharge  four  times  as  much  water  iu  the 
same  time. 

HULK. 

Square  the  given  diameter,  and  multiply  said  square 
by  the  given  proportion,  and  the  square  root  of  the  pro- 
duct is  the  answer. 

2J«2,5,  and  2,1x2,5=6,25  square. 

4  given  proportion* 

v/25,00=5  inch,  cliam.  .'7 


KVOLWTIOK,   OR  EXTRACTION   OF   ROOTS.          18S 

PROB.  VI.  The  sum  of  any  two  numbers,  and  their 
products  being  given,  to  find  each  number. 
RULE. 

From  the  square  of  their  sum,  subtract  4  times  their 
product,  and  extract  the  square  root  of  the  remainder, 
which  will  be  the  difference  of  the  two  numbers  ;  then 
half  the  said  difference  added  to  half  the  sum,  °;ives  the 
greater  of  the  two  numbers,  and  the  said  half  difference 
subtracted  from  the  half  sum,  gives  the  lesser  number. 

EXAMPLES. 

The  sum  of  two  numbers  is  43,  and  their  product  is 
442 ;  what  are  those  two  numbers  ? 

The  sum  of  the  numb.  43x43=1849  square  of  do. 

The  product  of  do.     442x  4=1768  4  times  the  pro. 

Then  to  the  4  sum  of  21,5  [numb. 

+and—  4,5  «/81=9  diff.  of  the 


Greatest  numb.  26,0 ")  4*  the  £  diff. 

E    S-  Answers. 

Least  numb.  17,0  J 

EXTACTION  OF  THE  CUBE  ROOT. 

A  Cube  is  any  number  multiplied  by  its  square. 

To  extract  the  cube  root,  is  to  find  a  number,  which, 
being  multiplied  into  its  square,  shall  produce  the  given 
number. 

RULE. 

1.  Separate  the  given  number  into  periods  of  three  fig- 
«res  each,  by  putting  a  point  over  the  unit  figure,  and 
•very  third  figure  from  the  place  of  units  to  the  left,  and 
if  there  be  decimals,  to  the  right. 

2.  Find  the  greatest  cube  in  the  left  hand  period,  and 
place  its  root  in  the  quotient. 

3.  Subtract  the  cube  thus  found,  from  the  said  period, 
and  to  the  remainder  bring  down  the  next  period,  calling 
this  the  dividend. 

4.  Multiply  the  suqare  of  the  quotient  by  SOO,  calling 
it  the  divisor. 


184         EVOLUTION,   OR    EXTRACTION   OF    ROOTS. 

5.  Seek  how  often  the  divisor  may  be  had  in  the  divi- 
dend, and  place  the  result  in  the  quotient ;  then  multiply 
4the  divisor  by  this  last  quotient  figure,  placing  the  pro- 
duct under  the  dividend. 

(..  Mu]t|ply  the  former  quotient  figure,  or  figures  by 
the  square  of  the  lust  quotient  figure,  and  that  product  by 
SO,  and  place  thf!  product  under  the  last ;  then  under  these 
two  products  place  the  cube  of  the  last  quotient  figure,  and 
add  them  together,  calling  their  sum  the  subtrahend. 

7.  Subtract  the  subtrahend  from  the  dividend,  and  to 
the  remainder  bring  down  the  next  period  fora  new  divi- 
dend ;  with  uhich  proceed  in  the  same  manner,  till  the 
whole  be  finished. 

NOTE. — If  the  subtrahend  (found  by  the  foregoing  rule) 
happens  to  be  greater  than  the  dividend,  and  consequent- 
ly cannot  be  subtracted  therefrom,  you  must  make  the 
last  quotient  figure  one  less;  with  which  find  a  ue'v  sub- 
trahend, (by  ttie  rule  foregoing)  and  so  on  until  you  can 
subtract  the  subtrahend  from  the  dividend. 

EXAMPLES. 

1.  Required  the  cube  root  of  18399,744. 

18599,744(26,4  Root.  Ant. 
8 

2x2=4x300  =  1200)10599  first  dividend.    0 

7200 

6x6=S6x2=r2xSO=2lGO 
6x6x6=  216 


9576  1st  subtrahend. 
26x26=676x300=202800)823744  2d  dividend. 

811200 

16x26=416x50=  12480 
G4 

823744  2d  subtrahend. 


BVOLUTIOtf,   OK   EXTRACTION   OF   ROOTS.          185 

NOTE. — The  foregoing  example  gives  a  perfect  root ; 
and  if,  when  all  the  periods  are  exhausted,  there  happens 
to  be  a  remainder,  you  may  annex  periods  of  cyphers,  and 
sontinue  the  operation  as  far  as  you  think  it  necessary. 

Answer.*. 

2.  What  is  th'e  cube  root  of  205579  ?  59 

S.  Of 614125  ?  85 

4.  Of  41421736?  346 

5.  Of         146363,183?  52,7 

6.  Ot        £9,503629  ?  5,09-f 

7.  Of        80,763  ?  4,52+ 

8.  Of ,162771356?  ,546 

9.  Of        ,000684134?  ,0884- 

10    Of 122615327232?  .  4968 

RULE  II. 

1.  Find  by  trial,  a  cube  near  to  the  given  number,  and. 
tall  it  the  supposed  cube. 

2.  Then  as  twice  the  supposed  cube,  added  to  the  giv- 
?n  number,  is  to  twice  the  given  number  added  to  the 
supposed  cube,  so  is  th»  root  of  the  supposed  cube,  to 
Lhe  true  root,  or  an  approximation  to  it. 

3.  By  taking  the  cube  of  the  root  thus  found,  forth* 
supposed  cube,  and  repeating  the  operation,  the  root  will 
>e  had  to  a  greater  degree  of  exactness. 

,  EXAMPLES. 

Let  it  be  required  to  extract  the  cube  rooWbf  2. 
Assume  1,3  as  the  root  of  the  nearest  cube;  then— 
!,3xl,3xl,3=2,l97=supposed  cube. 
Then,  2,197  2,000  given  number. 

£  2 

4,594  4,000 

2.000  2,197 


As  6,594      :      6,197    :    :    1,5     :     1,2599  root, 

ivhich  is  true  to  the  last  place  of  decimals  ;  but  might  by 

•epcating  the  operation,  be  brought  to  greater  exactness. 

-2.  What  5s  the  cube  root  of  584,arrW*  t 

c,  to- 


ISb          EVOLUTION,   OR   EXTRACTION  O?   ROOTS. 

5.  Required  the  cube  root  of  729001101  ? 

Jns.  900,0004 

QUESTIONS, 

Showing  the  use  of  the  Cube  Root. 

1.  The  statute  bushel  contains  2150,425  cubic  or  solid 
inches.     I  demand  the  side  of  a  cubic  box,  which  shall 
contain  that  quantity  ? 

.3/2150,425=12,907  inch.  Ans. 
Note. — The  solid  contents  of  similar  figures  are  in 
proportion  to  each  other,  as  the  cubes  of  their  similar 
sides  or  diamei^rs. 

2.  If  a  bullet  5  inches  diameter,  weigh  4lb.  what  will 
a  bullet  of  the  same  metal  weigh,  whose  diameter  is  6 
inches  ? 

3xSx3=«27  6x6x6=2i6  As  27  :  4lb.  :  :  216  : 
3211).'  Jlns. 

3.  If  a  solid  globe  of  silver,  of  3  inches  diameter,  be 
worth  150  dollars;  what  is  the  value  of  another  globe  of 
silver,  whose  diameter  is  six  inches  ?          g 

5x5x3=27  6x6x6=216  As  27  :  150  :  :  216  : 
SI 200.  Jlns. 

The  side  of  a  cube  being  given,  to  find  the  side  of  that 
cube  wich  shall  be  double,  triple,  &c.  in  quantity  to  the 
given  cube. 

RULE. 

Cube  your  £*/ven  side,  and  multiply  by  the  given  pro- 
portion between  the  given  and  required  cube,  and  the 
•ube  root  of  the  product  will  be  the  side  sought. 

4.  If  a  cube  of  silver,  whose  side  is  two  inches,  be  worth 
20  dollars ;  I  demand  the  side  of  a  cube  of  like  silver. 
'vhtse  value  shall  be  8  times  as  much  ? 

2x2x2=8  and  8x8=64^/64=4  incfes.  Jlns. 

5.  There  is  a  cubical  vessel,  whose  side  is  4  feet ;  I 
demand  the  side  of  another  cubical  vessel,  which  shall 
contain  4  times  as  much  ? 

4x4x4=64  and  64x4  =256 ^256«6,349-f-/t.  Ans. 

6.  /  cooper  having  a  cask  40  inches  long,  and  3:-'  in- 


EVOLUTION     OR    EXTRACTION   OF    ROOTS.  1&7 

chca  at  the  bung  diameter,  is  ordered  to  make  another 
cask  of  the  same  shape,  but  to  hold  just  twice  as  much  j 
what  will  be  the  bung  diameter  and  Length  of  the  new 
cask  ? 

40x40x40x2=128000  then  &  128000= 50,3 -f  length. 
32x32x32x2=65530  and  ^/G5536=40,3-f  bung  diam. 


»f  General-ltulefor  Extracting  the  Roots  of  ail  Poirers, 
RULE. 

1.  Prepare  the  given  number  for  extraction,  by  point 
ing  oft'  from  the  unit's  place,  as  the  required  root  directs 

2.  Find  the  first  figure  of  the  root  bv  trial,  and  subtract 
its  power  from  the  left  hand  period  of  the  given  number. 

3.  To  the  remainder  bring  down  the  first  figure  in  the 
next  period,  and  call  it  the  dividend. 

4.  Involve  the  root  to  the  next  inferior  power  to  that 
which  is  given,  and  multiply  it  by  the  number  denoting 
the  given  power,  for  a  divisor. 

5.  Find  how  many  times  the  divisor  may  be  had  iii 
the  dividend,  and  the  quotient  will  be  another  figure  of 
the  root. 

6.  Involve  the  whole  root  to  the  given  power,  and  sub- 
tract it  (always)  from  as  many  periods  of  the  given  num 
her  as  you  have  found  figures  in  the  root. 

7.  Bring  down  the  first  figure  of  the  next  period  to  the 
remainder  for  a  new  dividend,  to  which  fuid  a  new  divi- 
sor, as  before,  and  in  like  manner  proceed  till  the  whole 
be  finished. 

NOTE. — When  the  number  to  be  subtracted  is  greater 
than  those  periods  from  which  it  is  to  be  taken,  tne  la«t 
quotient  figure  must  be  taken  less,  &c. 

EXAMPLES. 

1.  llequired  the  cube  root  of  135796,744  by  the  ai>ov« 
•jent'ial  method. 


188          EVOLUTION,   OH   EXTRACTION   OF   ROOTS. 


135796744(5 1,4  the  root. 
125=lst  subtiaheml 

75)107  dividend. 

152651  =2d  subtrahend. 
7803)  31457=2d  dividend. 

1 35796744 =3d  subtrahend. 

5x5x3=75  first  divisor. 
51x51x51  =  132651  second  subtranend. 
51x51x3=7803  second  divisor. 
514x514x514=.135796744  third  subtrahend, 

0.  Required  the  sursolid,  or  fifth  root  of  6436343, 

6436343)23  root 
32 

2x2x2x2x5=80)323  dividend. 
23 X23 x23 x23 X23 =6436343  subtrahend^, 

NOTE. — The  roots  of  most  powers  may  be  found  by  the 
square  and  cube  roots  only ;  therefore,  when  any  even 
power  is  given,  the  easiest  method  will  be  (especially  in 
a  very  high  power)  to  extract  the  square  root  of  it,  which 
reduces  it  to  half  the  given  power,  then  the  squire  root  of 
that  power  reduces  it  to  half  the  same  power  ;  and  so  on, 
till  you  come  to  a  square  or  a  cube. 

tor  example:  suppose  a  12th  power  be  given:  the 
square  root  of  that  reduces  it  to  a  sixth  power  :  aud  the 
square  root  of  a  sixth  power  to  a  cube. 

KXAMM.V.S. 

3.  What  is  the  biqiiadrate,  or  4th  root  of  19987173376  .- 

Jns.  376. 

4.  Extract  the  square,  cubed,  or  6Ui  root  of  12230590 
464.  Ans.  48. 

5.  Extract  the  square,  biquadrate.  or  8th  root  of  721 SS 
95789388336.  Jlns.  96 


ALLIGATION  189 

ALLIGATION, 

IS  the  method  of  mixing  several  simples  of  different  qual- 
ities, so  that  the  composition  may  be  of  a  mean  or  middle 
quality  :  It  consists  of  two  kinds,  vi/,.  Alligation  Medial,, 
and  Alligation  Alternate. 

ALLIGATION  MEDIAL, 

Is  when  the  quantities  and  prices  of  Several  tilings  are 
given,  to  find  the  mean  price  of  the  mixture  composed  of 
those  materials. 

RULE. 

As  the  whole  composition  :  is  to  the  whole  value  :  :  so 
is  any  part  of  the  composition  :  to  its  mean  price. 

EXAMPLES. 

1.  A  farmer  mixed  15  bushels  of  rye,  at  64  cents  a 
bushel,  1 8  bushels  of  Indian  corn,  at  55  cts.  a  bushel,  and 
21  bushels  of  oats, at 28  cts.  a  bushel;  I  demand  what  a 
bushel  of  this  mixture  is  worth  ? 

bu.       eta.     facts.         bii.     g  cfs.         bu. 
15  at  64=9,60    As  54  :  25,38   :   :  1 
18        5.7=9,90  1 

21        28=5.88  cts. 

54)25,38(,47  'Answer. 
54  25,38 

2.  If  20  bushels  of  wheat  at  1  dol.  55  cts.  per  bushel, 
be  mixed  with  10  bushels  of  rye  at  90  cents  per  bushel, 
what  will  a  bushel  of  this  mixture  be  worth  ? 

Jns.  gl,  20cte. 

3.  A  Tobacconist  mixed  36  Ib.  of  Tobacco,  at  13.  6d. 

Ker  Ib.  12  Ib.  at  2s.  a  pound,  with  12  Ib.  at  Is.  lOd.  par 
). ;  what  is  the  price  of  a  pound  of  this  mixture  ? 

Jns.  Is.  8rf. 

4.  A  Grocer  mixed  2  0.  of  sugar,  at  56s.  per  C.  and  1 
C.  at  43s.  per  C.  and  2  C,'.  at.  50s.  per  C.  together;  I  de- 
mand the  j)rice  of  3  cwt.  of  this  mixture?  Jlns.  £7  13s. 

5.  A  "Wine  merchant  mixes  15  gallons  of  wine  at  4s. 
2d.  per  gallon,  with  24  gallons  at  6s.  8d.  and  20  gallons, 
at  6s.  3d. ;  what  is  a  gallon  of  this  composition  worth  ? 

Ms.  5s.  Wd. 


I'.'fl  ALLIGATION   ALTERNATE. 

t>.  A  grocer  lialh  several  sorts  of  su«ar,  viz.  one  son 
at  8  tlol.s.  percwt.  another  .sort  at  9  dots,  percwt.  a  third 
sen*,  at  10  do's,  ^or  cv.f.  and  a  fourth  sort  at  12  dols.  per 
cwt.  and  ho  •uould  mix  an  equal  quantity  of  each  togeth- 
er;  I  demand  the  price  of  Si  cwt.  of  this  mixtii' 

Jus.  £34    IZcts.  .Int. 

7.  -\  ')  molted  together  5  Ib.  of  silver  bullion, 

of  8  07..  line,  10  !b.  of  7oz.  fine,  and  15  Ib.  of  6  07..  line  ; 
pray  \vlwit  is  the  quality,  or  fineness  of  this  composition  r 

Jlns.  Goz.  ISpwt.  Sgi-.Jinc. 

Si  Suppose  5  Ib.  of  £old  of  £2  carats  fine,  2,  Ib.  of  21 
carats  fine,  and  I  Ib.  of  alloy  be  melted  together  ;  what  la 
the  quality,  or  fineness  of  this  mass  r 

19  carats  fine. 


ALLIGATION   ALTERNATE, 

IS  the  method  of  finding  what  quantity  of  each  of  the 
ingredients,  whose  rates  are  given,  will  compose  a  mix- 
ture of  a  given  rate ;  so  that  it  is  the  reverse  of  alligation 
ntedial,  and  may  be  proved  by  it. 

CASE.  I. 

"When  the  mean  rate  ot  the  whole  mixture,  and  the 
rate*  of  all  the  ingredients  are  given  without  any  limited 
quantity. 

RUI1J. 

1.  Place  the  several  rates,  or  prices  of  tho  simples,  be- 
in:1;  reduced  to  one  denoiad nation,  in  a  column  under  eath 
1  the  mean  price  in  the  like  name,  at  the  left 
kind. 

-2.  Connect,  or  link,  the  price  of  each  simple  or  ingre- 
dient, which  is  loss  than  that  of  the  mean  rate,  with  one 
or  any  number  of  those,  which  are  jp-eator  than  the  mean 
rate,  and  each  greater  rate,  or  price  with  one,  or  any 
number  of  the  less. 

3.  Place  the  difference,  between  the  mean  price  (or 
mixture  rate)  and  that  of  each  of  the  simples,  opposite 
to  ine  rotes  with  which  they  are  connected. 


ALLIGATION'     ALTERNATE.  191 

•4.  Then,  if  only  one  difference  stands  against  any  rate, 
it  will  be  the  quantity  belonging  to  that  rate,  but  it  there 
be  several.,  their  sum  will  be  the  quantity. 

EXAMPLES. 

1.  A  merchant  has  spices,  some  at  9d.  per  Ib.  some  at 
Is.  some  at  2s.  and  some  at  2s.  Gd.  per  Ib.  how  much  of 
each  sort  must  he  mix,  that  he  may  sell  the  mixture  at  Is. 
oil.  per  pound  ? 

(L        (I     Ib.  d.        Ib. 

9 ,10  at  9"|  f  «1        q   ^ 

4     12  \Gives  the  d.J  12-;-     10  i  £ 

8     24  f  Answer,  or  2(H  24 J       11  f  2 

11     30  j  1.30 — '     8J  "S 

2.  A  grocer  would  inix  the  following  quantities  of  su- 
jVti" ;  vi/..  at  10  cents,  13  cents,  and  1(1  cts.  per  Ib. ;  what 
'juantity  of  each  sort  must  be  taken  to  make  a  mixture 
v.-orth  fc  cents  per  pound  ? 

.ins.  5lb.at  Wets.  Qlb.  at  iScts.  and  2lb.  at  1C  els.  jier  Ib. 
5.  A  grocer  has  two  sorts  of  tea,  viz.  at  9s.  and  at  15s. 
per  Ib.  how  must  he  mix  them  so  us  to  afford  the  compo- 
sition for  12s.  per  Ib.? 

Ans.  He  must  mix  an  equal  quantity  of  each  sort. 

4.  A  goldsmith  would  mix  gold  of  17  carats  fine,  with 
<i.»Moof  19,  21,  and  24  carats  fine,  so  that  the  compound 
may  be  22  carats  fine;  what  quantity  of  each  must  he 
take. 

Ans.  2  nf  each  of  the  first  three  sorts,  and  9  of  the  last. 

5.  It  is  required  to  mix  several  sorts  of  rum.  vi/..  at  5s. 
Ts.  and  9s.  per  gallon,  with  water  at  0  per  gallon  to- 
rrthor,  so  that  the  mixture  may  be  worth  6s.  per  gallon  ; 
how  much  of  each  sort  must  the  mixture  consist  of? 

JHV.  1  gal.  of  Hum  at  5s.  1  do.  at  7s.  G  tin  at  9s.  and  5 
pals,  icafer.  Or,  3  gals,  rum,  (it  5s.  G  do.  at  7s.  1 
do.  at  9.1--.  and  I  gal.  u-fttcr." 

G.  A  grocer  hath  several  sorte  of  sugar,  vi/..  one   sort 

U  12  cts.  per  Ib.  another  at  11  cts.  a  third  at  9  cis.  and  a 

fourth  at  8  cts.  per  Ib. ;  I  demand  how  much  of  rach  sort 

;;;:;st  lie  mix  t»)geTher,  that  the  whole  quanti'v  maybe 

'  :tl  at  10  cents  per  pound  ? 


192  ALTERNATION    PARTIAL. 

lb.      cts.  lb.     cts.  tb. 

f2  at  12  fl  at  12  fS  at  12 

; 


.2  at    8  LI  at    8  jj  at   8 

4th  Ans.  5tb.  of  each  sort.* 

CASE  II. 

ALTERNATION  PARTIAL. 

f 

Or,  when  one  of  the  ingredients  is  limited  to  a  certain 
quantity,  thence  to  find  the  several  quantities  of  the  rest, 
in  proportion  to  the  quantity  given. 

RULE. 

Take  the  difference  between  each  price,  and  the  mean 
rate,  and  place  them  alternately  as  in  CASK  I.  Then,  as 
the  difference  standing  against  that  simple  whose  quantity 
is  given,  is  to  that  quantity  :  so  is  each  of  the  outer  dif- 
ferences, severally,  to  the  several  quantities  required. 

EXAMPLES. 

1.  A  farmer  would  mix  10  bushels  of  wheat,  at  TO  cts. 
per  bushel,  with  rye  at  48  cts.  corn  at  56  cts.  and  barley 
at  50  cts.  per  bushel,  BO  that  a  bushel  of  the  composition 
may  be  sold  for  58  cents;  what  quantity  of  each  must 
be  taken. 


Mean  rate,  3£< 

1 
As  «  :  10  :  :  - 

"70  ^  8  stands  against  the  given  nuan 
48-j      12                                         [tity 
S6J      1  10 
_SO  '  52 
"   2  :    2J  bushels  of  rye. 
10  :  12J  bushels  of  corn. 
52  :  40    bushels  of  barlev. 

.A                             '            

*  Thewfvur  answers  artftefromasinany  various  ways 
<\f  linking  the  rates  of  the  ingredients  together. 

Quest  ions  in  this  rule  adm  it  of  an  infinite  variety  nf  an- 
swers :  for  after  the  quantities  are  found  from  different 
methods  of  linking :  any  other  numbers  in  t,\e  same  proper '• 
t ion  between  themselves,  as  the  numbers  which  compost  tht 
fl"<-'rvr.  irill  U^ji-if  «/v.'/cfy  the  cnr'Hti'tw*  <>ftb<>  'ii'p^i^r. 


PARTIAL.  LU3 

2.  How  much  waier  must  be  mixed  with  100  gallons 
of  rum,  worth  7s.  6d.  per  gallon,  to  reduce  it  to  6s.  3d. 
per  gallon  ?  ./fos.  20  gallons. 

3.  A  fanner  would  mix  20  bushels  of  rye,  at  65  cents 
per  bushel,  with  barley  at  51  cts.  and  oats  at  30  cts.  per 
bushel;  how  much  barley  and  oats  must  be  mixed  with 
the  20  bushels  of  rye,  that  the  provender  may  be  worth 
41  cents  per  bushel  ? 

Jlns.  20  bushels  of  barley,  and  61T9T  bushels  of  oats. 

4.  With  95  gallons  of  rum  at  f>s.  per  gallon,  I  mixed 
ether  rum  at  6s.  8d.  per  gallon,  and  some  water;  then  1 
found  it  stood  me  in  6s.  4d.  per  gallon ;  I  demand  how 
much  rum  and  how  much  water  I  took  ? 

»flnj?.  95  gals,  rum  at  6s.  Sd.  and  50  gals,  renter. 

CASE  III. 

When  the  whole  composition  is  liiftited  to  a  given  quantify . 
RULE. 

Place  the  difference  between  the   mean  rate,  and  the- 
several  prices  alternately,  as  in  CASK  1. ;  then,  As  the 
sum  of  the  quantities,  or  difference,  thus  determined,  is  to 
the  given  quantity,  or  whole  composition  :  so  is  the  dirte- 
rcucc  of  each  rate,  to  the  required  quantity  of  each  rat.i.-. 

EXAMPLES. 

1.  A  grocer  had  four  sorts  of  tea,  at  Is.  ;1s.  fi*.  and  1G«. 
per  Ib.  the  worst  would  not,  sell,  and  the  best  we're  too 
dear;  he  therefore  mixed  120  h>.  and  so  much  tf  e.icli 
sort,  as  to  sell  it  at  4s.  p-:r  U>. ;  how  much  of  each  sort  di<J 
he  tako  ? 

Ib. 

f()  :  60  at  \] 

}  2  :  20  —  3! 

-    1  :  10  —  G  fpei 

:  30  —  \0  j 

1QQ     ' 


194  ARITHMETICAL    PROGRESSION. 

2.  How  much  water  at  0  per  gallon,  must  be  mixed 
with  wine  at  90  cents  per  gallon.  ?o  as  to  till  a  vessel  of  100 
gallons,  which  may  be  afforded  at  GO  cents  per  gallon  ?  j 

»'l»s.  f>.<U  gals,  u-uter.  and  66^  gals.  wine. 

3.  A  grocer  having  sugars  at  8  cts.  16  cts.  and  24  cts. 
per  pound,  would  make  a  composition  of  240  Ib.  worth 
20  cts.  per  Ib.  without  gain  or  Joss  ;  what  quantity  of  each 
must  be  taken  ? 

.ins.  40  Ib.  at  8  cts.  40  at  16  cts.  and  160  at  24  cts. 

4.  A  goldsmith  had  t\vo  sorts  of  silver  bullion,  one  of 
10  07..  and  the  other  of  5  o-/..  fine,  and  has  a  mind  to  mix 
a  pound  of  it  s;>  that  it  shall  be  8  oz  tine;  how  much  of 
each  sort  must  he  take  ? 

JMS.  4*  of  5  oz.f.n?,  and  7\  of  10  oz.fine. 

5.  Hramly  at  3s.  (Jil.  and  js.  Od.  per  gallon,  is  to  be 
mixed,  so  that  a  hhd.  o!  Cms  mm-  !)e  sold   for  12/. 
12j.  ;  how  many  gallons  must  I.e  taken  of  each  ? 

*i.'«s.  14    •«/??.  af  '5s.  (Jrf.  anc/  49    -a/s.  rti  5s.  6<f. 


A  R  ITH  M  ETIC  A  L  PROGRESSION. 


rank  of  numbers  more  than  two,  increasing  by 
common  excess,  or  decreasing  by  common  difference,  t» 
said  to  be  in  Arithmetical  Progression. 

<()  C  £,  4,  G,  8,  &c.  is  an  riscc'iiding  arithmetical  series: 

(  8,  (j.  4,  -2,  &r.  is  a  descending  arithmetical  series: 

The  numbers   which   form   t!  called  the 

terms  of  the  progression  ;  the  first  and  last  terms  of  which 

are  callqd  the  extrv;: 

PROBLEM  I. 

';rst  fenn,  tlie  last  term,  and  <l:c  number  of  terms 
loing  given,  to  fifid  the  SUM  i-l  .ill  the  terms. 


*.J  »ei-ii's  in  pr-i^r  JUT  futrts,  rlx.  the 

tyrmjlast  f  t,'rms,  common  difference, 

,-  an  ii  Hi  rve  of  ,  .-  nther  tiro 

y  !•,•  fuund,  which  ad;iut\  /  <>J'  Problems  :  but 
' 


ma 

'  r 


ARITHMETICAL    PROGRESSION.  195 

.       RULE. 

Multiply  the  sum  of  the  extremes  by  the  number  ot 
terms,  and  half  tho  product  will  be  the  answer. 

EXAMPLES. 

1.  The  first  twin  of  an  arithmetical  series  is  3,  the  last 
term  23,  and  the  number  of  terms  11 ;  required  the  sum 
of  the  series. 

23-j-3=25  sum  of  the  extremes. 
Then  26x11-4-2=143  the  Answer. 

2.  How  many  strokes  docs  the  hammer  of  a  clock 
strike,  in  twelve  hours  ?  .flns.  78 

S.  A  merchant  sold  100  yards  of  cloth,  viz.  the  first 
yard  for  1  ct.  the  second  for  2  cts.  the  third  for  3  cts.  &c. 
I  demand  what  the  cloth  came  to  at  that  rate  ? 

Ms.  8504. 

4.  A  man  bought  19  yards  of  linen  in  arithmetical  pro- 
gression, for  the  first  yard  he  gave  Is.  and  for  the  last  yd. 
I/.  17s.  what  did  the  whole  come  to?      Jlns.  £18  Is. 

5.  A  draper  sold  100  yds.  of  broadcloth,  at  5  cts.  for 
the  first  yard,  10  cts.  for  the  sec.md,  15  for  the  third,  &c. 
increasing  5  cents  for  every  yard  :  What  did  the  whole 
amount  to;  and  what  did  it  average  per  yard  ? 

Jlns.  Amount,  g252  },  and  the  average  price  is  g2, 52cfs. 
5  mills  per  yard. 

6.  Suppose  144  oranges  were  laid  2  yards  distant  from 
each  other,  in  a  right  line,  and  a  basket  placed  two  yards 
from  the  first  orange,  what  length  of  ground  will  that  boy 
travel  over,  who  gathers  them  up  singly,  returning  with 
them  one  by  one  to  the  basket  ? 

Ans.ZS  miles,  5  furlongs,  180  yds. 

PROBLEM  II. 

The  first  term,  the  last  term,  and  the  number  of  terete 
given,  to  find  the  common  difference. 

RULE. 

Divide  the  difference  of  the  extremes  by  the  number 
of  terms  less  1,  and  the  quotient  will  be  the  common  dif- 
ference. 


195  ARITHMETICAL    PROGRESSION. 

I 
EXAMPLES. 

1 .  The  extremes  are  3  and  29,  and  the  number  of 
terms  14,  what  is  the  common  difference  ? 

Extremes. 


Number  of  terms  less  1=13)26(2  .0ns. 

2.  A  man  had  9  sons,  whose  several  ages  differed  alike, 
the  youngest  was  3  years  old,  and  the  oldest  35;  what 
was  the  common  difference  of  Iheir  ages  ? 

Ans.  4  years. 

5.  A  man  is  to  travel  from  New-London  to  a  certain 
place  in  9  days,  and  to  go  but  3  miles  the  first  day,  in- 
creasing every  day  by  an  equal  excess,  so  that  the  last 
day's  journey  may  be  43  miles  :  Required  the  daily  in- 
crease, and  the  length  of  the  whole  journey  ? 

Jlns.  'Die  daily  increase  is  5,  and  the  "whole  journey 
£07  miles. 

4.  A  debt  is  to  be  discharged  at  16  differt^i.  nayments 
(in  arithmetical  progress! on.) the  first  payment  is  to  be 
141.  the  last  lOOJ. :  What  is  the  common  difference,  and 
the  sum  of  the  whole  debt  ? 

Jins.  5l.  14«.  8rf.  common  difference,  and  9121.  the  whole 
debt. 

PROBLEM  III. 

Given  the  first  term, last  term,  and  common  difference, to 
find  the  number  of  terms. 

RULE. 

Divide  the  difference  of  the  extremes  by  the  common 
difference,  and  the  quotient  increased  by  1  is  the  number 
of  terms. 

EXAMPLES. 

1.  If  the  extremes  be  3  and  45,  and  the  common  dif- 
ference 2 ;  what  is  the  number  of  terms  ?        Jlns.  22. 

2.  A  man  going  a  journey,  travelled  the  first  day  five 
miles,  the  last  day  *»  miles,  and  each  day  increased 
lii*  journey  by  4  miles}  how  many  days  did  he  travel, 
and  how  fur  ? 

Jus.  1 1  days,  and  tlie  whole  distance  travelled  275  mrks 


GEOMETRICAL    PROGRESSION.  197 

GEOMETRICAL  PROGRESSION, 

IS  when  any  rank  or  series  of  numbers  increased  by  one 
common  multiplier,  or  decreased  by  one  common  divisor  , 
as  1,  2,  4,  8,  16,  Sec.  increase  by  the  multiplier  2;  and 
£7,  9,  3,  1,  decrease  by  the  divisor  3. 

PROBLEM  I. 

The  first  term,  the  last  term  (or  the  extremes)  and  tht 
ratio  given,  to  find  the  sum  of  the  series. 

RULE. 

Multiply  the  last  term  by  the  ratio,  and  from  the  pro- 
duct subtract  the  first  term  ;  then  divide  the  remainder 
by  the  ratio,  less  by  l,and  the  quotient  will  be  the  sum 
oF  all  the  terms. 

EXAMPLES. 

1.  If  the  series  be  2,  6,  18,  54,  162,  486,  1458,  and 
the  ratio  3,  what  is  its  sum  total  ? 

3x1458—2 

: =2186  the  Answer. 

3—1 

2.  The  extremes  of  a  geometrical  series  are  1  and 
65536,  and  the  ratio  4;  what  is  the  sum  of  the  series? 

Atis.  87381. 

PROBLEM  II. 

Given  the  first  term,  and  the  ratio,  to  find  any  other  t?r<a 
assigned.* 

CASE  I. 

When  the  first  term  of  the  series  and  the  ratio  arc  equal. t 


*Jls  the  last  term  in  a  long  series  of  numbers  is  wry  te- 
dious to  be  found  by  continual  multiplications,  it  it- ill  be 
necessary  for  the  readier  finding  it  out,  to  have  a  series 
of  numbers  in  arithmetical  proportion,  called  indices, 
whose  common  difference  is  1. 

f  When  the  first  term  of  the  seriesttnd  the  ratio  are  equal, 
indices  must  begin  with  the  unit,  and  in  this  case,  ike 
17* 


(03  GEOMETRICAL    PROGRESSION. 

1.  Write  down  a  lew  of  the  leading  terras  of  the  se-^ 
r.ies,  and  place  their  indices  over  them,  beginning  the 
i.idices  with  an  unit  or  1. 

2.  Add  together  such  indices,  whose  sum  shall  make 
uji  the  entire  index  to  the  sum  required. 

5.  Multiply  1  he  terms  of  the  geometrical  series  belong- 
ing to  those  indices  together,  and  the  product  will  be  the 
term  sought. 

EXAMPLES. 

1.  Ifthe  first  be  2,  and  the  ratio  2;  what  is  the  13th 
term. 

1,2,.%    4,     5,  indices.  Then  5-f  5+3=13' 

2,  4,  8,  10,  32,  leading  terms.     32x32x8=8192  JJns. 

2.  A  draper  sold  £0  yards  of  superfine  cloth,  the  first 
yaid  fur  3d.  the  second  for  9d.  the  third  for  27d.  &c.  in 
triple  proportion  geometrical ;  what  did  the  cloth  come 
to  at  that  rate  ? 

The  °J  th,  or  last  term  is  3486784401  d. 
Then  3+3486784401—3 

=523017G600rf.  the  sum  of  all 

r»  _i 

tii*  terms  (by  Prob.  I.)  equal  to  £21792402  10s.  Jns. 

5.  A  rich  miser  thought  20  guineas  a  price  too  much 
for  12  fine  horses,  but  agreed  to  give  4  cents  for  the  first, 
1 6  cents  for  the  second,  and  64  cents  for  the  third  horse, 
and  so  on  in  quadruple  or  fourfold  proportio'n  to  thelastr 
\vhat  did  they  come  to  at  that  rate,  and  how  much  did 
they  test  per  head,  one  with  anuther  ? 

Arts.  Tlie  12  Worses  came  to  £225696,  20c*s.  and  tin 
Average  price  uas  £18641,  35cts.  per  head. 

product  nf  any  two  terms  is  equal  to  that  term,  signified 
by  the  sum  nf  their  indices. 

Tlius    5  I  ^  •*    4    5  ^Cc'  Indices  or  arithmetical  series 

'    ^  2  4  8  16  32  <jjj*c.  geometrical  series. 
AI.'".    3+2  =*    5  =  the  imle.v  of  the  ffth  term,  and 
4x8  —  32  «=  the  fifth  term 


GEOMETRICAL   PROGRESSION,  199 

CASE  II. 

AY  hen  the  first  term  of  the  series  and  the  'ratio  are  diffe- 

rent, that  is,  when  the  first  term  is  either  greater  or 

less  than  the  ratio.* 

1.  Write  down  a  few  of  the  leading  terms  of  the  series, 
and  begin  the  indices  with  a  cypher:  Thus,  0,  1,  2,  3,  &,c. 

-2.  Add  together  the  most  convenient  indices  to  mak« 
an  index  less  by  1  than  the  number  expressing  the  place 
of  the  term  sought. 

S.  Multiply  the  'terms  of  the  geometrical  series  to- 
gether belonging  to  those  indices,  and  make  the  product 
a  dividend. 

4.  Haise  the  first  term  to  a  pmvor  whose  index  is  one 
less  than  the  number  of  the  terms  multiplied,  and  make 
the  result  a  divisor. 

o.  Divide,  and  the  quotient  is  the  term  sought. 

EXAMPLES. 

4.  If  the  first  of  a  geometrical  series  be  4,  and  the  ratio 
3,  what  is  tine  7th  term  r 
0,     1,2,      3,  Indices. 
4,  12,  56,  108,  h>adin«;  terms. 

S  +  2+l*=6,  the  index  of  the  7th  term. 
108x36x12=40656 

--  =2916  the  7th  term  required. 

16 

Here  the  number  of  terms  multiplied  are  three;  there- 
fore the  first  term  raised  to  a  power  U:>s  than  three,  is  the 
2d  power  or  square  of  -4  =  16  the  divisor. 

*  When  the  first  term  of  the  scries  and  the  ratio  are  dif- 
ferent ..the  indices  must  begin  with  a  cypher,  and  the  sum 
of  the  indices  made  choice  of  i.iust  be  one  leas  than  the.  num- 
ber of  terms  given  in  the  question  :  because  I  in  the  indices 
stands  over  the  second  term,  and  £  in  the  indices  over  the. 
third  term,  <§'c.  a?;;l  in  this  caw,  the  product  of  amj  ftco 
terms,  divided  %  the  first,  is  equal  to  that  term  beyond  the 
first,  signified  by  the  sum  of  ikeir  indices. 

' 


T/i»«     5°'  !»  2'    S'     4' 

'    I  1,  3,  9,  27,  81,  cyr.  Geomet  ricnl  srrie.s. 

Here  4  +  3=7  the  inde.v  of  the  8th  term. 
81  x  27=21  37  the  8f/i  term,  or  the  7lh  beyond  the 


£00  vosmojr. 

5.  A  Goldsmith  sold  1  Ib.  of  gold,  at  2  cents  for  the 
first  ounce,  8  cents  for  the  second,  32  -cents  for  the  third, 
£c.  in  a  quadruple  proportion  geometrically;  what  did 
the  whole  come  to  ?  rfns.  gl  11848,  Wets. 

3.  "What  debt  can  be  discharged  in  a  year,  by  paying 
1  farthing  the  first  month,  10  farthing?,  (or  2^x1.)  the  se- 
cond, and  so  on,  each  mryitlrin  •*  ^nsfold  proportion  ? 
£  11 5740740  14s.  9d.  Sqrs. 

7.  A  thresher  worked  20  days  for  a  farmer,  and  receiv- 
ed fov  the  first  day's  work  four  barley-corns,  for  the  second 
1 2  barley -corns,  for  the  third  ?(5  barley-corns,  and  so  on 
in  triple  proportion  geometrical.    1  demand  what  the  20 
•  lays'  labor. came  to,  supposing  a  pint  of  barley  to  contain 
7680  corns,  and  the  whole  quantity  to  be  sold  at  2s.  6d. 
per  bushel  ?    »flns.  £1773  7s.  6d.  rejecting  remainders. 

8.  A  man  bought  a  horse,  and  by  agreement  was  to 
jjive  a  farthing  for  the  first  nail,  two  for  the  second,  four 
lor  the  third,  obc.    There  were  four  shoes,  and  eight  nails 
in  each  shoe;  what  did  the  horse  come  to  at  that  rate  ? 

Jns.  £4473924  5s.  Sjrf. 

9.  Suppose  a  certain  body,  put  in  motion,  should  move 
ilu-  length  of  one  barley-corn  the  first  second  of  time,  one 
inch  the  second,  ;uul  three  inches  the  third  second  of 
time,  and  so  continue  to  increase  its  motion  in  triple  pro- 
portion geometrical :  how  many  yards  would  the  said 
body  move  in  the  term  of  half  a  minute  ? 

.flns.   953199G85G23  yds.  }ft.  lin.  Ib.c.  irhich  is  no 
less  than  jive  hundrzd  and  forty-one  millions  of  miles. 

POSITION. 

JT  OS  IT  ION  is  a  rule  which,  by  false  or  supposed  num- 
bers, taken  at  pleasure,  discovers  the  true  ones  required. 
K  Is  divided  into  two  parts,  Single  or  Double. 

SINGLE  POSITION, 

Is  when  one  number  is  required,  the  properties  of 
which  are  given  in  the  question. 


SINGLE    rOSITION.  £('!] 

RULE. 

1.  Take  any  number  and  perform  the  same  operation 
with  it,  as  is  described  to  be  performed  in  the  question. 

2.  Tken  say;  as  the  result  of  the  operation  :  is  to  th« 
given  sum  in  the  question  :  :  so  is  the  supposed  number  : 
to  the  true  one  required. 

-The  method  of  proof  is  by  substituting  the  answer  in 
the  question. 

EXAMPLES. 

1.  A  schoolmaster  being  asked  how  many  scholars  he 
had,  said,  If  I  had  as  many  more  as  I  now  have,  half  as 
many,  one-third  and  one-fourth  as  many,  I  should  then 
have  148:  How  many  scholars  had  lie  ? 

Suppose  he  had  12  As  37  :  148  :  :  12  :  48  Ans 

as  many  =  12  48 

$  as  many  =6  24 

£  as  many  =4  16 

i  as  many  =    3  12 

Result,    37  Proof,  148 

2.  What  number  is  that  which  being  incrtased»by  A,  ^, 
aid  i  of  itself,  the  sum  will  be  125  ?' 

3.  Divide  93  dollars  between  A,  B  and  C.  so  that  B's 
share  may  be  half  as  much  as  A's,  andC's  ^::ire  three 
times  as  much  as  B's. 

•4ns.  A's  share  51,  B's  15$,  and  C's  46 J  dolls.. 

4.  A,  B  and  C,  joined  their  stock  and  gained  360  dols. 
of  which  A  took  up  a  certain  sum,  B  took  5  A  times  as 
much  as  A,  and  C  took  up  as  much  as  A  and  B  both ; 
vrhat  share  of  the  gain  had  each  ? 

Ans.  A  840,  B  g!40,  and  C  g!80. 

5.  Delivered  to  a  banker  a  certain  sum  of  money,  to 
receive  interest  for  the  same  at  61.  per  cent,  per  annum, 
simple  interest,  and  at  the  end  of  twelve  years  received 
7SH.  principal  and  interest  together  :  What  was  the  sum 

I  delivered  to  him  at  first  ?  Ans.  £425. 

6.  A  vessel  has  3  cocks,  A,  B  and  C  ;  A  can  lill  U  in 
1  hour.  B  in  2  hours,  and  C  in  4  hours  ;  in  what  time  wilt 
they  all  fill  it  together  ?  Ans.  34?nin.  I7±s«c. 


20hi  DOUBLE    POSITION 

DOUBLE  POSITION, 

AEACHES  to  resolve  questions  by  making  two  suppft- 
5itions  of  false  numbers.* 

RULE. 

f 

I.  Take  any  two  convenient  numbers,  and  proceed 
v.itli  each  according  to  the  conditions  of  the  question. 
Find  how  much  the  results  are  different  from  the 
Its  in  the  question. 

Multiply  the  first  position  by  the  last  error,  and  the 
Jast  position  by  the  first  error. 

4.  If  the  errors  are  alike,  divide  the  difference  of  the 
products  by  the  difference  of  the  errors,  and  the  quotient 
will  be  the  answer. 

j.  If  the  errors  are  unlike,  divide  the  sum  of  the  pro- 
ducts by  the  sum  of  the  errors,  and  the  quotient  will  be 
tlie  answer. 

NOTE.  The  errors  are  said  to  be  alike  when  they  are 
both  too  great,  or  both  too  small :  and  unlike,  when  one 
is  too  great,  and  the  other  too  small. 

EXAMPLES. 

1.  A  purse  of  100  dollars  is  to  be  divided  among  4 
men,  A,B,  C  and  D,  so  that  B  may  have  4  dollars  more 
than  A,  and  C  8  dollars  mo,  c    nan  B,  and  D  *wi» 
many  as  C  :  what  is  each  one's  share  of  the  money  ? 
1st.  Suppose  A    G  2(1.  Suppose  A    S 

B  10  B  12 

C  18  C  20 

D  5G  D  40 

70  80 

100  100 

1st.  error  2d.  error    20 

*  Those  question*,  in  ichich  the  results  tire  not  propor- 
tional to  their  position*,  belong  to  this  rule  ;  such  as  those, 
in  u'hich  the  number  sought  is  increased  or  diminished  by 
some  given  number,  which  is  no  known  part  of  the  number 
required 


DOUBLE    POSITION.  203 

The  errors  being  alike,  are  both  too  small,  therefore, 

Pos.         Err. 
6  SO 


X 


8  20 

240          120 
120 

10)120(12  A'spart. 

2.  A ,  Band  C, built  a  house  which  cost  500  dollars, 
ot  which  A  paid  a  certain  sum  ;  B  paid  10  dollars  more 
than  A,  and  C  paid  as  much  as  A  and  B  both  j  how  much 
urn  each  man  pay  ? 

Jins.  A  paid  120,  B  130,  and  C  250  dots. 

5.  A  man  bequeathed  100/.  to  three  of  his  friends,  afte, 
this  manner:  the  first  must  have  a  certain  portion :  the 
second  must  have  twice  as  much  as  the  first,  wantin'o-  SI. 
and  the  third  must  have  three  times  as  much  as  the  lust, 
wanting  151. :  I  demand  how  much  each  man  must  have  ? 

Jins.  TheJlr*t£2Q  lO.s.  second £33,  third  £46  10s. 

4.  A  laborer  was  hired  for  CO  days  upon  this  condition : 
that  for  every  day  he  wrought  he  should  receive  4s.  and 
tor  every  to  he  was  idle,  should  forfeit  2s.:  at  tho  ex- 
piration ot  the  time  he  received  7 1.  10s.;  hov/manvdav* 
aid  he  work,  and  how  many  was  he  idle  ? 

Jlns.  He  wrought  45  'days,  and  was  idle  15  r/ays. 

o.  What  number  is  that  which  being  increased  by  its 
$,  its  i,  and  18  more,  will  be  doubled  ?  Jlns.  72. 

6.  A  man  gave  to  his  three  sons  all  his  estate  in  monov, 
'17..  to  F  half,  wanting  501.  to  G  olie-thinl,  and  to  II  tfie 
est,  which  was  Wl.  less  than  the  share  of  G  ;  [demand 
•fie  sum  »ivon,and  each  man's  part? 

ins.  Thesnm  given  ivas  £3GO,  whereof  Fhad  /ISO. 


204  PKaMUTATlON    OF    QUANTITIES. 

7.  Two  men,  A  and  B,  lay  out  equal  sums  of  money 
in  trade  ;  A  gains  126£.  and  B  looses  8?/.  and  A's  money 
is  now  double  to  IVs  :  \\hat  did  each  lay  out  ? 

Jilts.  £500. 

8.  A  farmer  having  driven  his  cattle  to  market,  reciv- 
ed  for  them  all  ISO/,  being  paid  for  every  ox  71.  for  every 
cow  5l.  and   for  every  calf  I/.  10s.  there  were  twice  as 
many  cows  as  oxen,  and  throe  times  as  many  calves  as 
eows  ;  how  many  were  there  of  each  sort  ? 

Jlns.  5  oxen,  10  cows,  and  30  calves. 

9.  A,  B  and  C,  playing  at  cards,  staked  524  crowns  ; 
but  disputing  about  tricks,  each  man  took  as  many  as  he 
could  :  A  got  a  certain  number ;  B  as  many  as  A  and  15 
more  ;  C  got  a  fifth  part  of  both  their  sums  added  togeth- 
er :  how  many  did  each  get  ? 

Jlns.  A  127J,  B  142},  C  54. 


PERMUTATION  OF  QUANTITIES, 

IS  the  showing  how  many  different  ways  any  given 
number  of  things  may  be  changed. 

To  find  the  number  of  Permutations  or  changes,  that 
can  be  made  of  any  given  number  of  things,  all  different 
from  each  other. 

RULE. 

Multiply  all  the  terms  of  the  natural  series  of  numbers, 
from  one  up  to  the  given  number,  continually  together 
and  the, last  product  will  be  the  answer  required. 


EXAMPLES. 


1.  How  many  changes  can  be 
made  of  the  three  first  letters  u£ 
the  alphabet?  Proof; 


Ans. 


a  b  c 
a  c  b 
b  a  c 
b  c  a 
c  b  a 
cab 


3.  IIww  maTiv  changes  may  be  rung  en  9  bells? 

362S80. 


ANNUITIES   OR    PENSIONS.  £05 

3. » Seven  gentlemen  met  at  an  inn,  ami  \vcrc  so  well 
pleased  with  their  host,  and  with  each  other,  that  they 
agreed  to  tarry  so  long  as  they,  together  with  their  host, 
could  sit  every  day  in  a  different  position  at  dinner  ;  how 
long  must  they  have  staid  at  said  inn  to  have  i'u Hilled 
their  agreement  :  dits.  llO^j  year*. 


AXNUITIKS  OR   PKNSIONS, 

COMPCTKD    AT 

COMl}0  U.VI)  LVTEREST. 
("ASK   I. 

To  find  the  amount  of  an  annuity,  or  Pension,  in  arrears, 
at  Compound  Interest. 

RULE. 

1.  Make  1  the  first  term  of  a  geometrical  progression, 
and  the  amount  of  gl  or  £  1  tor  one  year,  at  the  given 
rate  p«r  cent,  the  ratio. 

2.  Carry  on  the  series  up  to  as  many  terms  as  the  given 
number  of  years*,  and  find  its  sum. 

3.  Multiply  the  sum  thus  found,  by  the  given  annuity, 
and  the  product  will  be  the  amount  sought. 

KX  \MIT.KS. 

1.  If  125  duls.  vearly  rent,  or  annuity,  be  forborne,  (or 
unpaid)  4  years;  what  will  it  amount  to,  at  G  per  cent, 
per  annum,  compound  interest? 

1  +  1,06+1, 1236+1, l910lG=4.Sr46lG  sum  of  the 
scries**— Then,  4,374Cloxl2o=S54G,827  the  amount 
sought. 

OR  BY  TABLK   I 

Multiply  tMe  Tabular  number  under  the  rate  and  op- 
posite to  the  time,  by  the  annuity,  and  the  product  will  be 
the  amount  sought. 

*Tlte  sum  of  the  series  thus  found,  is  the  ancnntt  of 
\l.  or  1  dollar  annuit;,'.  far  the  given  time,  iruich  mat/  be 
fuuntlin  Table.  JL  ready  calculated. 

Hence,  either  the  a  mount  or  present  worth  of  nnnuitiea 
m.7y  be  readi!:/ found  by  Tables  for  that 


206  ANN  CITIES   OR    PENSIONS. 

2.  If  a  salary  of  GO  dollars  per  annum  to  be  paid  year- 
ly,  be  forborne  '20  years,  at  0  per  cent,  compound  in- 
terest j  what  is  the  amount  ? 

Under  6  per  cent,  and  opposite  20,  in  Table  II,  you 
will  find, 
Tabular  number  =36,78559 

60  Annuity. 

-9ns.  S2207,13540=g:220r,  IScts.  5?n.+  . 

5.  Suppose  an  Annuity  of  100J.  be  12  years  in  arrears, 
it  is  required  to  find  what  is  now  due,  compound  interest 
being  allowed  at  5/.  per  cent,  per  annum  ? 

.iiis.  £1591  i4s.  3,024rf.  (by  Table    II.) 

4.  What  will  a  pension  of  120Z.  per  annum,  payable 
yearly,  amount  to  in  3  years,  at  5/.  per  cent,  compound 
interest?  .ins.  £578  6s. 

II.  To  find  the  present  worth  of  Annuities  at  Compound 
Interest. 

RULE. 

Divide  tl>e  annuity,  &c.  by  that  power  of  the  ratio  sig- 
nified by  the  number  of  years,  ana  subtract  the  quotient 
from  the  annuity:  This  remainder  being  divided  by  the 
ratio  less  1,  the  quotient  will  be  Iho  present  value  of  the 
Annuity  sought. 

EXAMPLES.' 

1.  Whatready  money  will  purchase  an  Annuity  of  50J. 
to  continue  4  years,  at  5l.  per  cent,  compound  interest? 


=-1,215506)50,00000(41,13513  + 

From  50 

Subtract        41,13513 


ii.'  1  ,05—1  «05)  8.86487 


ANNUITIES   OR   TENSIONS.  £07 

BY  TABLE  II 

Under  5  per  cent,  and  even  with  4  years, 
We  have  3,54595  =present  worth  of  ll.  for  4  years. 
Multiply  by          50=Annuity. 

*flns.  £177,29750=present  worth  of  the  annuity. 

2.  What  is  the  present  worth  of  an  annuity  of  60  dols. 
per  annum,  to  continue  20  years,  at  6  per  cent,  compound 
interest?  Jitis.  g688  J9i cts.+ 

3.  What  is  SOZ.  per  annum,  to  continue  7  years,  worth 
in  ready  money,  at  6  per  cent,  compound  interest  ? 

Jlns.  £167  9s.  5d.  +  - 

III.  To  find  the  present  worth  of  Annuities,  Leases,  &c- 
taken  in  REVERSION,  at  Compound  Interest? 

1.  Divide  the  Annuity  hv  that  power  of  the  ratio  deno- 
ted by  the  time  of  its  continuance. 

2.  Subtract  the  quotient  from  the  Annuity :  Divide  the 
remainder  by  the  ratio  less  I,  and  the  quotient  will  be  the 
present  worth  to  commence  "immediately. 

3.  Divide  this  quotient  by  that  power  of  the  ratio  deno- 
ted by  the  time  of  Reversion,  (or  the  time  to  come  before 
the  Annuity  commences)  and  the  quotient  will  be  the 
present  worth  of  the  Annuity  in  Reversion. 

EXAMPLES. 

1.  What  ready  money  will  purchase  an  Annuity  of  501. 
payable  yearly,  for  4  years  :  but  not  to  commence  till  two 
years,  at  5  per  cent.  ? 

4th  power  of  1,05=1,215506)50,00000(41,13513 
Subtract  the  ,quotient=41, 13513 

Divide  by  1,05— 1  =,05)8,86487 
£d.  power  01  1,05=1, IQ25}177,297(160.8136=£1GQ 
16s.  Sd.  1  jr.  present  worth  of  the  Annuity  in  Reversion. 

OR  BY  TABLE  III. 

Find  the  present  value  of  \L  at  tfie  given  rate  for  the 
sum  of  the  time  of  continuance,  and  time  in  reversion 
added  together ;  from  which  value  subtract  the  present 
worth  of  I/,  for  the  time  in  reversion,  and  multiply  the  re- 
mainder bv  the  Annuity ;  the  product  will  be  the  answer. 


808  ANNUITIES   OR    FKXS1ONS. 

Tims  in  Example  1. 
Thv.c  of  continuance,  4  years. 
Ditto  of  reversion,        2 

The  sum,                  =6  years,  gives  5.075695 
Time  in  reversion,    =2  years,  1,859410 


Remainder,      3,216282x50 

.flns.  £160.8141 

2.  \Vliat  is  the  present  worth  of  75l.  yearly  rent,  which 
is  not  to  commence  until  10  years  hence,  and  then  to  con- 
tinue 7  years  after  that  time  at  6  per  cent.  ? 

.'Ins.  £235  15s.  9rf. 

5.  What  is  the  present  worth  of  the  reversion  of  a 
lease  of  GO  dollars  per  annum,  to  continue  £0  years,  but 
not  to  commence  till  the  end  of  8  years,  allowing  G  per 
cent,  to  the  purchaser?  Jns.  S4S1  78efs.  ~fV"' 

IV.  To  find  the  present  worth  of  a  Freehold  Estate,  or 
an  Annuity  to  continue  forever,  at  Compound  Interest. 

HULK. 

As  the  rate  per  cent,  is  to  100/.  :  so  is  the  yearly  rent  to 
the  value  required.  EXAMPLES. 

1 .  What  is  the  worth  of  a  Freehold  Estate  of  40/.  per 
annum,  allowing  5  per  cent,  to  the  purchaser  ? 

As  £5  :  £100  :  :  £40  :  £800  Ans. 

2.  An  estate  brings  in  yearly  150/.  what  would  it  sell 
for,  allowing  the  purchaser  G  per  cent,  for  his  money? 

Jns.  £2500 

V.  To  find  the  present  worth  of  a  Freehold  Estate,  HI 

Reversion,  at  Compound  Interest. 

RULE. 

1.  Find  the  present  value  of  the  estate  (by  the  fore^n- 
iii2;  riih  }  as  though  it  were  to  be  entered  on  immediately, 
and  di  ii.e  the  said  value  by  that  power  of  the  ratio  de- 
noted by  the  time  of  reversion,  iirul  the  quotient  will  be 
the  present  worth  of  the  estate  in  Kevei 

KXAMl'I 

1.  Suppose  a  freehold  estate  of  4l)f.  per  annum  to  com- 
mence two  \ears  hence,  be  put  on  sale ;  what  is  its  value, 
allowing  the  purchaser  5l.  j;er  ceut.  r 


qUESTIOKS   FOR     EXERCISE,      •  £09 

As  5  :  100  :  :  40  :  800 =prcsent  worth  if  entered  on 
immediately. 

Then,  1,05  =  1.1025)800,00(725.62S58=725/.  12s. 
5irf.=present  worth  of  £800  in  two  years  reversion.  Ans. 

OR  BY  TABLE  III. 

Find  the  present  worth  of  the  annuity,  or  rent,  for  the 
time  of  reversion,  which  subtract  from  the  value  of  the 
immediate  possession,  and  you  will  have  the  value  of  the 
estate  in  reversion. 

Thus  in  the  foregoing  example, 
l,859410=present  worth  of  ]/.  for 2  years. 
40=annuity  or  rent. 

74,376400  =present  worth  of  the  annuity  or  rent,  for 

[the  time  of  reversion. 

From  800,0000 =value  of  immediate  possession. 
Take     74,3764  =present  worth  of  rent. 


£725,6236=/;725  12.«.  5Arf. 

2.  Suppose  an  estate  of  90  dollars  per  annum,  to  com 
mence  10  years  hence,  were  to  be  sold,  allowing  the  pur- 
chaser 6  per  cent. ;  wiuit  is  it  worth  ? 

..  $837,  59c/.s.  2m. 

3.  "Which  is  the  most  advanlr.  -cms,  a  term  of  15  years, 
in  an  estate  of  IQQl.  per  annun  ;  or  the  reversion  of  susb 
an  estate  forever  after  the  «;iiu  i  5  years,  computing  at  the 
rate  of  5  per  cent.  JVM- annum,  cr-n^ound  interest? 

.5ns.  The  first  term  of  lo  years  is  better  than  the  re- 
vgrsiou  forever  afterwards,  by  /*75  18s.  7 id, 

'       Jv  /w 


A  COLLECTION  OF  QUESTIONS  TO  EXHRCISB 
THE  FOREGOING  RULES. 

1.  I  demand  the  sum  of  1748£  added  to  itsolf  ? 

Jlns.  3497. 

2.  What  is  the  difference  between  41  eagles,  and  4099 
dimes  ?  .fltzs.  lOcfs. 

3.  What  number  is  that  which  being  multiplied  by  21^ 
the  product  will  be  1365  ?  Aus.  65. 

19* 


$10  QUESTIONS    FOR     KXURCISE. 

A.  What  number  is  that  which  being  divided  by  1,9,  the 
quotient  \vill  bo  7-2  :  Jlns."  1363. 

5.  What  number  is  that  which  being  multiplied  by  !§, 
the  product  \vil!  be  £  r  Jns.  ^. 

(>.  There  are  7  chests  of  drawers,  in  each  of  which 
there  arc  IK  drawers,  and  in  each  of  these  there  are  six 
divisions,  in  each  of  which  is  161.  6s.  8d.;  how  much 
money  is  there  in  fhe  whole  ?  Jns.  £12348. 

7.  Nought  SG  pipes  of  wine  for  453G  dollars :  h»w  must 
I  sell  it  a  pipe  to  sue  one  for  my  own  use,  and  sell  the. 
rest  for  what  the  whole  cost?  Ans.  S129,  GOc/s. 

8.  Just  16  yards  of  German  serge, 
For  90  dimes  had  1 ; 

How  many  vards  of  that  same  cloth 

Will  14  eagles  buy  ?     Jus.  248yJs.  Sqrs.  2fna. 

9.  A  certain  quantity  of  pasture  will  last  963  sheep  7 
weeks,  how  many  must  be  turned  out  that  it  will  last  the 
remainder  9  weeks?  llns.  214. 

10.  A  grocer  bought  an  equal  quantity  of  sugar,  tea, 
and  coffee,  for  740  dollars;  he  gave  10  cents  per  Ib.  for 
the  sugar,  60  cts.  per  Ib.  for  the  tea.  and  £0  cts.  per  Ib.  for 
the  coneej  required  the  quantity  of  each? 

.4ns.  83d*.  f>or.  8{jrfr. 

11.  Bought  cloth  at  glj  a  yard,  and  lost  2.5  per  cent. 
how  was  it  sold  a  yard  ?  .#?/>.  '.'.i  ;<•/.<,•. 

12.  The  third  part  of  an  army  was-  killed,  the  fourth 
part  taken  prisoners,  and  1000  lied;  how  many  were  in 
this  army,  how  many  killed,  and  how  manv  captives  r 

Jus.  2400  in  the  annii,  800  HUM,  ami 
600  taken  pritnurrs. 

13.  Thomas  sold  150  pine  apjiif--  Qts  a  pi<ve. 
and  received  as  much  money  as  Ham    i<'iL-i\cd  for  u 
certain   number  of  water-nu'lloiis.  uKirli  he  sold   at  25 
cents  a  piece:  how  imich  monoy  <\\\]  rjirli  vfc<-:'. P.  ?;iul 
Unv  many  mellons  havl  liar. 

Ans.  Each  received  goO,  and  llnn-.j  sold  200  mellnns. 

14.  Said  John  to  Dick,  my  purse  and  money  arc  worth 
9/.  2s.  but  the  money  is  1  \\i-nty-live  times  as  much  us  the 
purse ;  I  demand  how  much  money  was  in  it  ? 

8  155. 


FOR    EXERCISE.  £11 

V 

15.  A  young  man  received  210?.  which  was  |  ef  lii» 
elder  brother's  portion  ;  now  three  times  the  elder  broth- 
er's portion  was  half  the  father's  estate ;  what  was  the 
value  of  the  estate  ?  dns.  £  1890. 

16.  A  liaro  starts  40  yards  before  a  greyhound,  and  it- 
not  perceived  by  him  till  she  has  been  up  40  seconds  ;  she 
sends  away  at  the  rate  often  miles  an  hour,  and  the  dog* 
on  view,  makes  after  her  at  the  rate  of  18  miles  an  how  : 
How  long  will  the  course  hold,  and  what  space  will  be  ran 
over,  from  the  spot  where  the  dog  started  ? 

./?;j.s.  G02\.sec.  and  530i/f/s.  space. 

17.  "What  number  multiplied  by  57  will  produce  just 
what  134  multiplied  by  71  will  do  ?  Ms.  iGCfi 

18.  There  are  two  numbers,  whose  product  is  1610,  the 
greater  is  given  46 ;  I  demand  the  sum  of  their  squares, 
and  the  cutye  of  their  difference  ? 

rfns.  The  sum  of  their  squares  is  5341.     Tlie.  cube  of 
their  difference  is  1331. 

19.  Suppose  there  is  a  mast  erected,  so  that  £  of  its 
Ungth  stands  in  the  ground,  1 2  feet  of  it  in  the  water,  and 
3  af  its  length  in  the  air,  or  above  water;  I  demand  the 
whole  length  ?  .fl?is.  216  feet. 

20.  What  difference  is  there  between  the  interest  of 
500/.  at  5  per  cent,  for  12  years,  and  the  discount  of  the 
same  sum,  at  the  same  rate,  and  for  the  same  time  ? 

itfns.  £112  IDs. 

21.  A  stationer  sold  quills  ai  11s.  per  thousand,  by 
which  he  cleared  |  of  the  money,  but  growing  scarce, 
raised  them  to  136.  6d.  per  thousand ;  what  might  he  clear 
per  cent,  by  the  latter  price  ? 

jfns.  £96  7s.  3TJTd 

22.  Three  persons  purchase  a  West-India  sloop,  to- 
wards the  payment  of  which  A  advanced  |,  B  ^,  and  C 
T40/.     How  much  paid  A  and  B,  and  what  part  of  th? 
vessel  had  C  ? 

Jlns.  Jlpnid  £267TST,  K  £30j75T,  and   C's  part  of 

the  1'w.el  u-as  ^. 

25.  What  is  the  purchase  of  1200/.  bank  stock,  at  103| 
percent?  »frts.  £1243  10f. 

24.  Bought  27  pieces  of  Nankeens,  cask  Hi  jtrds,  et 


$}£  QUESTIONS   FOR   EXERCISE. 

14s.  4£d.  a  piece,  which  were  sold  at  18d.  a  yard ;  re- 
quired the  prime  cost,  what  it  sold  for,  and  the  gain. 

£•   «•    *• 
C  Prime,  cosf,    19    8     1$ 

4*M  Sold  /or,       23    5     9 
(.  Gain,  3  17     Ti 

25.  Three  partners,  A.  B  and  C,  join  their  stock,  and 
buy  goods  to  tlie  amount  of  £1025,5  ;  (if  which  A  put  in 
a  certain  simi ;  J>  put  in. ...I  know  iu.i  lio\v  much,  and  C 
the  rest :  Vhey  gained  .it  the  rate  of  2-»/.  per  cent. :  A's 
part  of  th«.  gain  is  {,  B's  A,  and  C's  the  re^t.     Required 
each  man*!  particular  stock.  £. 

{Ws  stack  was  5 12  JS 
/r.s- __  205.1 
6"5 S07,C5 

26.  AVhat  is  1'iat  number  M hich  being  di^ ided  by  ^,  the 
quotient  will  be  21  ?  *dns.  15$. 

27     If  to  my  a;j;e  there  added  be, 

One-half.  <me-t.hird?  and  three  times  fliree, 
feix  score  and  ten  the  sum  v.ill  be; 
AViiat  is  rny  a^e.  pray  shew  jt  me  ? 

.!?j?s.  66. 

28.  A  gentleman  divided  his  fortune  among  his  three 
sons,  givin'.;  \  '.  /.  as  often  as  B  5/.  and  to  C  but  3/.  as 
often  as  B ''/('.  ;I.M!  yet  C's  dividend  was  2584/. ;  -whatdid 
the  whole  c.4;itn  rasiount  to  P 

Ans.  £19466  2s.  8d. 

29.  A  gentleanah  left  his  son  a  foHune,  i  of  which  he 
spent  in  three  mon(!-s ;  ^  of  the  remainder  lasted  him  10 
months  longer,  when  he  had  only  2524  dollars  left;  pray 
what  did  his  father  bequeath  him? 

Jlns.  S5889,  3Scte.-f 

30.  In  an  orchard  of  fruit  trees,  4  of  them  bear  apples, 
i  pea;-',  ^  ]»l;;ms,  4  )  of  ihem  peaches,  and  10  cherries; 
how  many  tioes  docs  the  orchard  contain  ?     .flns.  600, 

31.  There  is  a  certain  number,  which  beinjr  divided  by 
7.  i\\>~-  '|i!oticnt  rc^u'iin^  multiplied  by  3,  that  product 
iiivided  by  5,  I'rom  tlic  quotient  20  being  subtracted,  and 
SO  added'fo  the  remainder,  the  half  sum  shall  make  65  j 

•  -II  me  the  number  ?  ••/?»$.  14Q£>. 


QUESTIONS  FOB.  EXERCISE.  SIS 

32.  What  part  of  25  is  |  of  an  unit  ? 

Jlns.  ^V 

33.  If  A  can  do  a  piece  of  work  alone  in  10  days,  B  in 
20  days,  C  in  40  days,  and  D  in  80  days ;  set  all  four 
about  it  together,  in  what  time  will  they  finish  it  ? 

Jlns.  5-J-  days. 

34.  A  farmer  being  asked  how  many  sheep  he  had,  an- 
swered, that  he  had  them  in  five  fields,  in  the  first  he  had 
i  of  his  flock,  in  the  second  -J-,  in  the  third  |,  in  the  fourth 
«\,  and  in  the  fifth  450;  how  many  had  he  P 

Jlns.  1200. 

35.  A  and  B  together  can  build  a  boat  in  18  days,  and 
with  the  assistance  of  C  they  can  do  it  in  1 1  days ;  in 
what  time  would  C  do  it  alone  ?  Jins.  28f-  days. 

36.  There  are  three  numbers,  23,  25-,  and  42  ;  what  is 
the  difference  between  the  sum  of  the  squares  of  the  first 
and  last,  and  the  cube  of  the  middlemost  ? 

Jlns.  13332. 

57.  Part  1200  acres  of  land  among  A,  B,  and  C,*  so 
that  B  may  have  100  more  than  A,  and  C  64  more  than 
B.  Jlns.  Jl  S12,  B  412,  C  476. 

38.  If  3  dozen  pairs  of  gloves  be  equal  in  value  to  2  pie 
ces  of  Holland,  3  pieces  of  holland  to  7  yards  of  satin,  6 
yards  of  satin  to  2  pieces  of  Flanders  lace,  and  3  pieces  of 
Flanders  lace  to  81  shillings ;  how  many  dozen  pairs  of 
gloves  may  be  bought  for  28s.  ? 

Jlns.  2  dozen  pairs. 

S9.  A  lets  B  have  a  hogshead  of  sugar  of  18  cwt.  worth 
5  dollars,  for  7  dollars  the  cwt.  $  of  which  he  is  to  pay  in 
cash.  B  hath  paper  worth  2  dollars  per  ream,  which  he 
gives  A  for  the  rest  of  his  sugar,  at  2J  dollars  per  ream. — 
Which  gained  most  by  the  bargain  ? 

Jins.  Ji  by  g!9,  90cte. 

40.  A  father  left  his  two  sons  (the  one  1 1  and  tht  other 
16  years  old)  10000  dollars,  to  be  divided  so  that  each 
share,  being  put  to  interest  at  5  per  cent,  might  amount 
to  equal  sums  when  they  would  be  respectively  21  yean 
of  age.    Required  the  shares  ? 

Jlns.  5454^-  and  4545T»T  dollars. 

41.  Bought  a  certain  quantity  of  broadcloth  for  383? 


•14  QUESTIONS  ro:i  .EXERCISE. 

5=.  and  if  the  num  -lings  which  it  cost  per  yard 

were  added  to  the  ,  yards  bought,  the  sum  would 

be  586 ;  i  oer  of  yards  bought,  and  at 

what  price  r*»r  ;,  :uu  r 

.foj.  SG5j/rfs.  at  21.-;.  per  yard. 
Solved  by  PROBLEM  VI.  page  183. 
42.  TV/-.)  partner;";.  Peter  and  .luhn,  bought  goods  to  the 
amount  of  iOOO  dollars;  in  the  purchase  of  whi«h,  Peter 
paid  more  than  John,  and  John  paid. ...I  know  not  how 
much  :  They  i  their  goods  lor  ready  money,  and 

thereby  gained  at  »f  200  per  cent,  on  the  prime 

cost:  t!'.-y  (ii-  1-k-u  ihc  ^n\n  between  them  in  proportion 
to  the  purcha.ii-  i.vjney  '  .  aid  in  buying  the  goods ; 

and  Pet<-i  John,  My  part  of  the  gain  is  really  a 

handsome  sum  of  money  ;  i  wish  I  hod  a.s  many  such  sums 
as  your  part  contains  dollars,  I  should  then  have  £4)60000. 
1  demand  each  man's  particular  stock  in  purchasing  tho 
goods. 

dns.  Peter  paid  GOO  dollars,  and  Joint  paid  400. 

TKTC    FOLLOWING    QUESTIONS    ARE    PROPOSED    TO 
SURVEYORS. 

1.  Heqiv.rcd  to  lay  ort  a  lot  of  land  in  form  of  a  long 
square,  containing  3  a-.re.s,  2  rood.",  and  29  rods,  that  shall 
take  just  100  rods  of  wall  i •>  enclose,  or  fence  it  round  j 
prav  how  mar.y  roda  in  length,  and  how  many  wide,  must 
saiu  lot  be  ? 

Jin.;,  r-l  rr,  '.-;  in  /.';.•;;•'''•'.  (')'d  19  in  breadth. 

:\eu  !>y  PUOHI.KM  V.I.  pag^  ' 
tract  of  !:.nd  is  (•:»  be  iait!  o:if  in  foi  .n  of  an  equaJ 
squr:  .  ("lice.  5 

i' fence  slijili  contain  10  : 
IIov/  lar^e   n:u>t  t  .  square  \K- 

many  acres  us  tliercarc  rails  in  tin-  fi-:  icloses  it. 

so  tliat  e\  .  nco  a:i  a« 

..  Me  fra;  ';rad 

co.1;/ 

Thus,  J    mile=G20  rods:  then   3S20Xo:K»-:-l60r=.640 
acres  :  and  320x4x10  =  12800  rails.     AsG40  :  12SO 
12800  :  2JGOOO  rails,  which  will  enclose  £JGOOO  acreg=^ 
20  miles 


•    AN 

APPENDIX, 

CONTAINING 

SHORT  RULES, 
FOR  CASTING  INTEREST  AND  REBATE? 

TOGETHER   WITH    SOME 

USEFUL  RULES, 

7OR   FINDING   THE    CONTENTS  OF   SUPERFICIES,  SOLIDSj 
&C. 


SHORT  RULES, 
FOR  CASTING  INTEREST  AT  SIX  PER  CENT. 

I.   To  find  the  interest  of  any  sum  of  shillings  for  any 
number  of  days  less  than  a  month,  at  6  per  cent. 

HULK!' 

1.  Multiply  the  shillings  of  the  principal  by  the  num- 
ber of  days,  and  that  product  by  2,  and  cut  off  three 
figures  to  the  right  hand,  and  all  above  three  figures  will 
be  the  interest  in  pence. 

2.  Multiply  the  figures  cut  off  by  4,  still  striking  off 
three  figures  to  the  right  hand,  and  you  will  have  the 
farthings,  very  nearly. 

EXAMPLES. 

1.  Required  the  interest  of  51.  8s.  for  25  days. 
£.      «.. 

5,8=108x25x2=5,400,  and  400x4=1,600 

Jlns.  5d. 

2.  \Vhatistheinterestof  £1J.  Ss.  for 29  days? 

Ants,  2s.  Od. 


APPENDIX- 

FEDERAL  MONEY. 

U.  To  find  the  interest  of  any  number  nf  cents  for  any 
number  of  days  less  than  a  month,  at  0  per  cent. 

RULE. 

Multiply  the  cents  by  the  number  of  days,  divide  the 
product  by  6,  and  point  ott'  two  figure s  to  the  rie;ht.  and 
ail  the  figures  at  the  left  hand  of  the  dash,  will  be  the 
mterest  in  mills,  nearly. 

EXAMPLES. 

Required  the  interest  of  85  dollars,  for  GO  days. 
g       cts.  mills. 

85=8500x20-i-Ga=283,33  Jns.  283  which  is 

28cis.  3  HU//S. 

2.  What  is  the  interest  of  75  dollars  41  cents,  or  73-11 
cents,  for  27  days,  at  6  per  cent.  ? 

330  TJU//S,  or  33c/s. 


III.  When  the  principal  is«given  in  pounds,  shillings,  &c. 
New-England  currency,  to  find  the  interest  for  any 
number  of  days,  less  than  a  month,  in  Federal  Money. 

RULE, 

Multiply  the  shillings  "m  the  principal  by  the  number 
of  days,  and  divide  the  product  by  36,  the  quotient  win 
be  the  interest  in  mills,  for  the  given  time,  nearly;  omit- 
ting fractions. 

EXAMPLE. 

Required  the  interest,  in  Federal  Money,  of  27/.  15?. 
for  27  days,  at  (i  per  cent. 
£.    s.      s. 
27     15=5y5x27^-3Gr=4lCmi^s.=r41cKGjH. 


IV.  When  the  principal  is  jiven  in  Federal  Money,  and 
jou  want  the  interest  in  shillings,  pence,  &c.  New-Eng- 
land currency,  for  any  number  oj"  days  less  fhan  a 

" 


sir 

RULE. 

Multiply  the  principal,  in  cents, by  the  number  of  day^ 
and  point  oft'  five  figures  to  the  right  hand  of  the  product, 
which  will  give  the  interest  for  the  given  time,  in  shil- 
lings and  decimals  of  a  shilling,  very  nearly. 

EXAMPLES. 

A  note  for  65  dollars,  31  cents,  has  been  OH  interest  25 
days ;  how  much  is  the  interest  thereof,  in  New-England 
currency  ? 

$5  cts.  s.  5.  d.qrs* 

Ans.  65,3 1=653 1x25=1,63975=1  '7  2  " 

REMARKS. — In  thfc  above,  and  likewise  in  the  preced- 
ing practical  Rules,  (page  127)  the  interest  is  confined  at 
six  per  cent,  which  admits  of  a  variety  of  short  methods 
of  casting :  and  when  the  rate  of  interest  i»  7  per  cent,  as 
established  in  New-York,  &c.  you  may  first  cast  the  in- 
terest at d  per  cent,  and  add  thereto  one  sixtn  of  itself) 
and  the  sum  will  1  the  interest  at  7  per  cent,  which  per- 
haps, many  times,  will  be  found  more  convenient  than  the 
general  rule  of  casting  interest. 

EXAMPLE. 

Required  the  interest  of  75t.  for  5  months  at  7  per 
cent.  s. 

7,5  for  1  month. 
5 

37,5=1  17  6  for  5  months  at  6  per  cent. 
+  1**      63 

Jlns.  £2    39  for  ditto  at  7  per  ce^. 

A   SHORT   METHOD   FOR    FINDING  THE  REBATE    OF  Arfft 

GlVF.N    SUM,   FOR    MONTHS   ANQ    DATS. 

RULE. 

Diminish  the  interest  of  the  given  sum  for  the  time  by 
its  own  interest,  and  this  gives  the  Rebate  very  nearly. 

EXAMPLES. 

1.  Wbat  is  the  rebate  of  50  dollars  for  sii  month*,  at 
3  per  eenk  ?  1 9 


118 

'  S  cffi 

The  interest  of  50  dollars  for  6  moaths,  it         1    50 
And,  the  interest  of  1  dol.  50  cts.  for  6  months,  is       4 

•Int.  Itetate,  81    46 

£.  "What  is  the  rebate  of  150J.  for  7  months,  at  5  per 
cent.  ? 

Interest  of  150J.  for  7  months,  is         476 
Interest  of  41.  7s.  Gd.  for  7  mouths,  is       2    6* 

Jlns.  £4    4  Hi  nearly. 

By  the  above  Rule,  those  who  use  interest  tables  IB 
their  counting-houses,  have  onlv  to  deduct  the  interest  ot* 
the  interest,  and  tlie  remainder  is  the  discount. 

Jl  concise  Rule  to  reduce  the  cur  r  end -s  of  the  different 
State*,  tckere  a  dollar  is  an  even  number  of  sfiitlir^gSf 
to  Federal  Money. 

RULE  I. 

Bring  the  given  jum  into  a  decimal  expression  by  in 
jyection,  (as  m  Pi-ob!<Mn  I.  page  87)  then  uivide  the  \\ho!e 
by  ,S  in  New-Kii^land  and  by  ,4'in  New-York  currency, 
and  the  qaotienr  \vili  be  dollars,  cente,  &.c/ 

EXAMPLES. 

1.  Reduce  54l.  Ss.  5 Ad.  New-England  currency,  to 
Federal  Money. 

,5)54,415  decimally  expressed. 

Jns.  8181. S3  ft*. 

?,  Reduce  7s.  ll^d.  New-England  currency,  to  F<tvi«- 
ral  Mou 

75.  11  $.?.  =^0,399  then,  ,S),5G9 

J/IS.    ?: 

5.  Fcducr  513/.  15s.  lOd.  New-York,  &.«.  eurrcrcr, 
to  Federal  M 

,nal 


APPENDIX. 


4.  Reduce  10s.  5^d.  New-York,  &e.  currency,  to-  Fede- 

ral money. 

,4)0,974  decimal  of  19s. 


5.  Ucduce  G4?.  New-England  currency,  to  tedeval 
Mouev. 

,3)64000  decimal  expression. 

821  3.33  £  .flns. 

^SOTF..—  I5v  the  foregoing  rule  you  may  carry  on  the 
decimal  to  anv  degree  nf  exactness;  but  in  ordinary  i>rac- 
«be,  the  following  Contraction  may  L>e  useful. 

RULE  II. 

To  t'le  shillings  contained  in  the  given  sum,  annex  * 
times  the  given  pence,  increasing  the  product  'r;  'Z  }  t:ira- 
divide  the  whole  by  the  number  of  shillings  contained  in 
a  dollar,  and  the  quotient  will  be  cents. 

F.XAMPLK5. 

1.  Reduce  45s.  Gd.  >V,\  -England  currency,  to  Fede- 
ral Money. 

6x8-f2  =»  50  to  be  annexed. 
'  6)45,50    or     6)1550 

£7,58!  .9ns.       T53  f^Jif*.  =' 

2.  Reduce  C/.  10s.  Ud.  New-Yoj;k,  &.c.  currency,  to 
federal  Money. 

9x8+2=74  to  be  annexed. 

Then  8)5074  Or  thus,  8)^,74 

• 


,J»w.      634  ce??fs.=6  34  26,34 

N.  B.  When  there  are  no  pence  in  the  given  sum,  you 
must  annex  two  cyphers  to  the  shillings  ;  theu  divide  as 
before,  £c. 

3.  Reduce  31.  5s.  New-England  currency,  to  Federal 
Money. 

Si  5s.=>65s.     Theu  6)6500 

1U86  ce*U.' 


S20  APPENDIX. 

SOMtf  USEFUL  RULES, 

TOE    T1NDING    THE    CONTENTS    OJ   SUPERFICIES     AND 
SOLIDS. 

SECTION  I.    OF  SUPERFICIES. 

The  superficies  or  area  of  any  plane  surface,  is  com- 
posed or  made  up  of  squares,  either  greater  or  legs,  ac- 
cording; to  the  different  measures  by  which  the  dimen- 
sions or  the  figure  are  taken  or  measured : — and  because 
12  inches  in  length  make  1  foot  of  long  measure,  there- 
fore, 12x12^=144,  the  square  inches  in  a  superficial  foot, 
&c. 

ART.  I.  To  find  the  area  of  a  square  having  equal 
sides. 

RULE. 

Multiply  the  side  of  the  square  into  itself,  and  the  pro- 
duct will  be  the  area,  or  content. 
EXAMPLES. 

1.  How  many  square  feet  of  boards  are  contained  m 
the  floor  of  a  room  which  is  20  feet  square  ? 

20x20=400  feet,  the  Answer. 

2.  Suppose  a  square  lot  of  land  measures  26  rods  on 
each  side,  how  many  acres  doth  it  contain  ? 

NOTE. — 160  square  rods  make  an  were. 

Therefore,  26x9-f»=-676  sq.  rods,  and  676-r-l60=4a. 

S6r.  the  Answer. 

ART.  2.    To  measure  a  Parallelogram,  or  long  square. 
RULE. 

Multiply  the  length  bv  the  breadth,  and  the  product 
be  tne  area  or  superficial  content. 
EXAMPLES. 

1.  A  certain  garden,  in  form  of  along  square,  is  96  ft. 
long,  and  54  wide ;  how  many  square  feet  of  ground  are 
contained  in  it?  Jns.  96x54=5184  square  feet. 

2.  A  lot  of  land,  in  form  of  a  long  square,  is  120  rod? 
in  length,  and  60  rods  wide ;  how  many  acres  are  in  it  ? 

120x50=J200  sq.  rods,  tiien,  7T»gV=^5  acres,  Ans. 

3.  If  a  board  or  plank  be  21  feet  long,  and  18  inches 
eroad  ;  how  many  square  feet  are  contained  in  it  ? 

18  inches^  1,5  feet,  <tew21xl,5=»Sl,5  Ans. 


Or,  in  measuring  bo»*i!s,  you  may  multiply  the  length 
iu  feet  by  the  breadth  fn  inches,  and  divide  by  12,  the 
•uotieot  will  gke  the  answer  in  square  feet.  &.c. 

Thus,  in  t!ie  foregoing  example,  21X18---12— 31,5  as 
before. 

4.  If  a  board  be  S  inches  wide,  how  much  in  length 
will  make  a  square  foot  ? 

RULE.— Divide  144  by  the  breadth,  thus,    8)144 

»fl/is.  18  in.  ' 

5.  If  a  piece  of  land  be  5  rods  widQ,  how  many  rods  in 
length  will  make  an  acre  ? 

RULE.— Divide  160  bj$he  breadth,  and  the  quotient 
will  be  the  length  required,  thus,  5)160 

rfns.     32  rods  in  length. 
ART.  5.    To  measure  a  Triangle. 

Definition. — A  Triangle  is  any  three  cornered  figure 
vhicbis  bounded  by  three  right  lines.* 

RULE. 

Multiply  the  base  of  the  given  triangle  into  half  its 
perpendicular  height,  or  half  the  base  into  the  whole  per- 
pendicular, and  the  product  will  be  the  area. 

EXAMPLES. 

1.  Required  the  area  of  a  triangle  whose  base  or  long- 
vst  side  is  32  inches,  and  the  perpendicular  height  14 
inches.  32x7=224  square  inches,  the  Answer. 

2.  There  is  a  triangular  or  thre?  cornered  lot  of  land 
\vhose  base  or  longest  side  is  5H  rods  ;  the  perpendicular 
from  the  corner  opposite  the  base,  measures  44  rods ;  how 
many  acres  doth  it  contain  ? 

51,5x22=1153  square  rods,=*7  acres,  13  rods. 

*Jt  Triangle  may  be  either  right  tinglzd  or  oblique;  in 
either  case  the  teacher  can  easily  give  the  scholar  a  rigid 
idea  of  the  base  and  perpendicular,  by  marking  it  down 
on  a  state,  paper,  <§"c . 


TO  MEASURE  A  CIRCLE. 

ART.  4.  The  diameter  of  a  Circle  being  giren,  to 
find  the  Circumference. 

RULE. 

As  I* '  is  to  22  :  :  so  is  the  given  diameter  :  to  the 
circumference.     Or,  more  exactly,  As  113  :  is  to  355  • : 
&c.  the  diameter  is  found  inversely. 
.  NOTE.'— The  diameter  is  a  riglrf  line  drawn  across  the 
sircle  through  its  centre. 

EXAMPLES. 

1.  What  is  the  circumference  of  a  wheel  whose  diam- 
eter is  4  feet?— As  -7  :  22  :  :  4  :  12,57  the  circumfe 
rence. 

£.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is  S5r — As  7  :  22  :  :  55  :  110.5ns. — and  inversely 
as  22  :  7  :  :  110  :  55,  the  diameter,  &c. 

ART.  5.    To  find  the  area  of  a  Circle. 

RULE. 

Multiply  half  the  diameter  by  half  the  circumference, 
and  the  product  is  the  area ;  or  if  the  diameter  is  {riven 
without  the  circumference,  multiply  the  square  of  the 
diaioatgf  by  ,7854  and  the  product  w\\  be  the  area. 
EXAMPLES. 

1.  Required  the  area  cf  a  circle  whose  diameter  is  12 
inchos,  and  circamfereace  S7,7  ii;ches. 

1 8,85  =half  the  circumference. 
6=half  the  diameter. 

113,10  area  in  square  inches.. 

2.  Required  the  area  of  a  circular  garden  \\hiuc  diame- 
ter is  11  rods  ?  -)4 

By  the  second  method,  11x11  =  121 

9.1,0334  rods. 
SECTION  £.    OF  SOLU 

.>HJs  are  estimalfuj  by  the  solid  inch,  solid  ibyt,  &c. 
.8  of  Uiese  incKesTthat  «  12x12x12  make  I  oihic 
'  olid  foot. 


APPENDIX.  22S 

ART.  6.    To  measure  a  Cube. 

Definition. — A  cube  is  a  solid  of  six  equal  sides,  each 
of  \vhich  is  an  exact  square. 

RULE. 

Multiply  the  side  by  itself,  and  that  product  by  the 
same  side,  and  this  last  product  will  be  the  solid  content 
«>f  the  cube. 

EXAMPLES. 

•     1.  The  side  of  a  cubic  block  being  18  inches,  or  1  foct 
and  G  inches,  how  many  solid  inches  doth  it  contain  ? 
ft.  in.   ft. 

1  6=1,5  and  1,5x1,5x1,5=3,575  solid  feet ,  «3ns. 
Or,  18x18x18=5832  solid  inches,  and  {fff  =3,375. 
-2.  Suppose  a  cellar  to  be  dug  that  shall  contain  12  feet 
every  way,  in  length,  breadth  and  depth ;  how  many  solid 
feet  of  earth  must  be  taken  out  to  complete  the  same  ? 
12x12x12=1728  solid  feet,  tte  Answer. 

ART.  7.  To  find  the  content  of  any  regular  solid  of  three 
dimensions,  length,  breadth  and  thickness,  as  a  piece  ot 
timber  squared,  whose  length  is  more  than  the  breadth 
and  depth. 

RULE. 
Multiply  the  breadth  by  the  depth  or  thickness,  and 

that  product  by  the  length,  which  gives  the  solid  content. 

EXAMPLES. 

1.  A  square  piece  of  timber,  b<>i;iu;  I  !;><>!  *3  inches,  or 
18  inches  broad,  9  inches  thick,  and  \>  1'ivt  ur  108  inches 
leng:  how  many  solid  feet  doth  it  contain  : 

1  ft.    Gin.  =  1,5    foot. 

9  inches    =  .75  foot. 


Trod.    1,125x9=10,185  solid  fed,  the  AK  . 
in.  in.  in.    solid  in. 
Or,  18x9xl08=17496-M728=10,125  feet. 

But,  in  measuring  timber,  you  may  multiply  the  bread  tk 
in  inches,  and  the  depth  in  inches,  and  that  product  by 
the  length  in  feet,  and  divide  the  last  product  by  144, 
which  will  srive  the  salidfcontcnt  in  feet,  £*". 


Sfi4 

A 


A  piece  of  timber  being  16  inches  broad,  li  inches 
thick,  and  20  feet  long,  to  fuid  the  center*  ? 
Breadth  16  indie. 
Depth      11 

Prod.     176x20=5520    then,  S52C+  144  =<24,4  /<?<;£, 


S.  A  niece  of  timber  15  inches  bread,  8  wiches  thick, 
and  25  feet  loi;g  j  how  manv  solid  feet  doth  it  contain  ? 

.flns.  20,8  -f/af. 

ART.  8.  AVhen  the  breadtli  and  thickness  of  a  piece  of 
timber  are  given  in  inches,  to  find  how  much  in  length 
•will  make  a  bolid  foot. 

,  RULE. 

Divide  1728  by  the  product  of  the  breadth  and  depth, 
and  the  quotient  will  be  the  length  making  a  solid  loot. 
EXAMPLES. 

1.  If  a  piece  of  timber  be  11  inches  broad  and  8  inches 
deep,  how  majiy  inches  in  length  will  make  a  solid  foot  ? 

11X8=88)17£8(19,6  inches,  .Ins. 

2.  If  TI  piece  of  timber  be  IS  inches  broad  and  14  in- 
ches deep,  how  many  inches  in  length  will  make  a  solid 
foot  ? 

18x14=252  divisor,  then  252)172S(6,8  inches,  Jrw. 

ART.  9.    To  measure  a  Cylinder. 
ll-'ftnitinn.  —  A  Cylinder  is  m  round  body  whose  baics 
are  circles,  like  a  i  ountl  column  or  stick  of  timber,  of 
equal  bigness  from  end  to  end. 

j  TUTLE. 

Multiply  the  square  of  the  diameter  of  the  end  bjr 

,7«5-l  -.•hi'"!i  ?;ivcs  i;he  area  of  tl  ••   na^f  ;  then  i-inltiplv 
tiie  area  of  tNe  uase  bj  .1,  uu«I  the  product  \\ili 

be  the  solid  content.  " 

EXAMPI  K. 

"\VIiat  is  the  solid  centent  of  a  round  stick  of  timber  of 

whose  diameter  M  18  i?* 


equal  bignes*  from  end  to  end, 
clics,  and  Icrjgth  CO  feet  ? 


APPENDIX. 


225 


18  in.=l,5  ft. 
Xl,5 

Square  2,25  x,  7854  =1,767  15    area  of  the  base. 
'  X20  length. 


35,34300  solid  content. 
Or,    18  inches. 
18  inches. 

324x,7854=254,4fi%    inches  area  of  the  base. 
20  length  in  feet. 

144]5089,3920(35,34S  solid  fat,  JJns. 
ART.  10.     lo  find  how  many  solid  feet  a  round  stick  of 
timber,  eouall*  thick  from  end  to  end,  will  contain 
when  hewn  square. 

RULE. 

Multiply  twice  the  square  of  its  semi-diameter  in  in- 
ches by  the  length  in  feet,  then  divide  the  product  by  144 
and  tne  quotient  will  be  the  answer. 

EXAMPLE. 

If  the  diameter  of  a  round  stick  of  timber  be  22  inches 
and  its  length  20  feet,  how  many  solid  feet  will  it  contam 
when  hewn  square  ? 

11x11x2x20-144=33,6+  feet,  the  solidity  when 
hewn  square. 

ART.  11.  To  find  how  many  feet  of  square  edged  boards 
ot  a  given  thickness,  can  be  sawn  from  a  log  of  a  "ivea 
diameter. 

RULE. 

Bind  the  solid  content  of  the  log,  when  made  square, 
by  the  last  article—  Then  say,  As  the  thickness  of  the 
board  including  the  saw  calf  :  is  to  the  solid  feet  :  :  so  is 
12  (inches)  to  the  number  of  feet  of  boards. 

EXAMPLE. 

How  many  feet  of  square  edged  boards,  U  inch  thick, 
including  the  saw  calf,  can  be  sawn  from  a  log  20  fe«t 
long  and  24  inches  diameter  P 

l£xl2x2x20-:-144=40/^,  solid  content. 
As  U  :  40  :  :  12  :  384  feet, 


S26  APPENDIX. 

ART.  12.    The  length, breadth  :'.nd  d-ji>t!i  <.r a:iy  square 
box  being  given,  to  find  how  nu.ij  uu^iieis  a  \vtii  cuutatu. 

ROLE. 

Multiply  the  length  by  the  breadth,  and  thru 
by  the  depth,,  divide  the  last  jvcc'i-ici  !>y 
•olid  inches  in  a  statute  bushel,  and  tiie  <;'iotiejat  will  be  ' 
the  answer. 

KXAMIM  K. 

There  is  a  square  box,  the  I  ugth  of  .n  is  5 

inches,  breadth  of  ditto  -10  inci.e*.  a.ui  if,s  di-pth   is  60 
inches  ;  how  many  bushejs  of  cMru  will  it  ).,;>',<!  ? 

50x40x00-:- 21 50.425  =55,84+  or  5c»  bushels,  three, 
pecks.  JJus. 

ART.  15.  The  dimensions  of  the  vral'ts  of  a  brick  build- 
ing being  givn,  to  find  how  many  Bricks  are  nece«- 
sarv  to  build  it. 

RULE. 

From  the,  \tfhole  circumference  \>f  the  wall  measured 
round  on  the  outside,  subtract  f'rj  r  times  its  thickness, 
then  multiply  the  remainder  by  t  b  c  height,  and  that  pro- 
duct bv  t\  e  thickness  of  the  wa'J4.  jjives  the  solid  content 
of  t.he.*v,«hole  wall;  which  mo^Jphed  by  the  number  uf 
bncka  contained  in  a  solid  foo/.,  give*  the  ans«  cr. 


How  man/  bricks  8  inc'n  es  lon»,  4  inches  wide,  ami 
2$  inches  thick,  will  it  tak.c  to  build  a  house  44  feet  long, 
40'  fee1  wide,  and  20  fe«st.  high,  and  tke  walls  to  be  one 
foot  t' 

8x4x2,5=80  solid    inches  in  a  brick,  then  175 
c=21.6  bricks  in  .1  sn'm'.  foot. 

44_i_40-J-44-f40=i7.bS  fret,  \\hulo  leiij  '  *  of  wall, 
— 4  four  times  the  thickn*. 

H-4  remains. 
Multiply;' by     20  height. 

3280  solid  feet  in  the  whove  wa& 
Multiply  by     21,6  bricks  in  a  solid  foot. 

Product,    70848  bricks.  .1ns. 


APPENDIX.  £37 

AXT.  14.    To  find  the  tonnage  of  a  ship. 
RULE. 

Multiply  the  length  of  the  keel  by  the  breadth  of  toe 
beam,  and  that  product  by  the  depth  of  the  hold,  and  di- 
vide the  last  product  by  95,  and  the  quotient  ii  the  t«n- 
Mge. 

EXAMPLE. 

Suppose  a  ship  72  feet  by  the  keel,  and  24  feet  by  the 
beam,  and  12  feet  deep  ;  wkat  i.t  the  tonnage  ? 

72x24xl3-f-P5=«=2l8,2-fto«s.  Jns. 

KULK  II. 

Multiply  the  length  o&  the  keel  by  the  breadth  of  the 
beam,  and  that  product  by  hulf  the  breadth  »f  the  bean, 
and  divide  by  9j. 

EX. \MPI.E. 

A  ship  84  feet  by  the  keel,  28  feet  by  the  beam ;  what 
is  the  tonnage  ? 

84x28xl4-r-95=350.29  tons.  Jn.«. 

ART.  15.    From  the   proof  of  any  cabl*,   to  fuid  the 
strength  of  another. 

RULE. 

The  strenii;tl\  of  cables,  aiul  consequently  the  weights 
ef  their  anchors,  are  as  the  cube  of  their  peripheries. 
Therefore;  As  the  cube  of  the  periphery  ol  any  cable, 
Is  to  the  weight  of  ita  anchor  ; 
So  is  iho  cuijcof  tli e  periphery  of  any  other  cabl», 
To  tlie  weight  of  its  anchor. 
EXAMVI.ES. 

1.  If  a  cable  6  inches  about,  require  an  anchor  of  2$ 
€wt.  of  what  vrii^ht  niust  an  anchor  be  lor  a  1 2  inch  cable  ? 

As  6xfixfi  :  -2}cirt.  :  :  ls2xl2xH  :   .  ins. 

2.  If  a  12  iiich  cable  ref|uir-e  an  anchor  of  IS  c\vt.  what 
must  the  circumference  oi'  a  cable  be,  for  an  anchor  of 
Si  cwt.  ? 

ctt-t.  cwf.  in. 

As  18  :  12x12x12  :  :  2,2J  :  2If.v^!f- 

ART.  Ifi.     Having  the  dimensions  of  t,vo  similar  built 
ship*  of  a  different  tipacitv,  with  the  burthen  ' 
fcf  Uietw,  to  find  tl'.t;  burthen  of  the  ot!:cr 


2£3  APFEXDIX. 

RULE. 

The  burthens  of  similar  built  ships  are  to  each  other, 
as  the  cubes  of  their  like  dimensions. 

EXAMPLE. 

If  a  ship  of  500  tons  burthen  be  75  feet  long  in  the  keel, 
I  demand  the  burthen  of  another  ship,  whose  keel  is  100 
feet  long  ?  T.cwt.grs.lb. 

A»  75x75x75  :  500  :  :  100x100x100  :  711  2    0    24-f 


DUODECIMALS, 

OR 

CROSS  MULTIPLICATION, 

IS  a  rule  made  use  of  l>y  workmen  and  artificers  in  ea§t* 
ing  up  the  contents  of  th^ir  work. 

RULE. 

1.  Under  the  nvilti;)lr.;nd  .vrite  the  correspondingde' 
Bonunations  of  the  multip'Vr. 

2.  Multiply  each  len.:  into  <i.'»  nultiplicard.  beginning 
at  the  lowest,  by  the  lu^iK'st  deiv>-iiintion  in  the"  multi- 
plier, and  write  the  result  of  each  undtr  its  respective 
term  ;  observing  to  carry  an  uuk  ibr  eveiy  12,  from  each 
lower  denomination  to  ;  -nerior. 

3.  In  the  same  mannf    :    -It,   'y  ail  the  multiplicand 
fay  the  inches,  or  second  denomination,  in  tlic  multiplier, 
and  set  th«  n  -  v!i  term  on-.-  nlace  removed  to  the 
right  band  of  those  in  the  ini.ltipliriitn^. 

4.  Do  the  same  with  the  seconds  in  the  multiplier,  let- 
ting  the  result  of  each  term  two  places  to  the  right  hand 
•f  those  in  the  multiplicand.  &c. 

EXAMPLES. 

F.    /.  F.    I.  F.    I. 

Multiply    73  75  4    f 

15y  47  39  58 

29    0    "       27    9    9      25     6          91  10    1 
429 

Product,   .13     2    'J 


AWKNDIX.  ££9 


tF.   I 

'  Multiply      4    7 
By        '  r  5  10 
______   0 

Product,     26    8  10 

F.   /. 

3     8 
•  7    6 

27    6 

[F.    /• 

9     7 

S     6 

r  _ 

32    6 

-»«------._--• 

F.    /. 

7  10 

8  n 

Multiply 
By 

F.    /. 

3  11 
9    5 

"fTI           ~f 

6    5 

7    6 

Product,     36  10    7        48     1    6        69  10    2 


FEET,   INCHES   AND   SECONDS. 

F.   /.    " 

Multiply     986 
By  793 

[tiplier. 

67  1 1    6   '"       =prod.  by  the  feet  in  the  mut- 
734    6N""=dittoby  the  inches. 
251    6=ditto  by  the  seconds. 


75     5    3 

7    6  j? 

F.    /.    " 

567 
8    9  10 

F.    1.    " 

7    1    9 

Z    8    9 

w     "// 

Multiply 
By 

Product,  55    2939        481128   10 


How -many  square  feet  in  a  board  16  feet  9  iuchct 
long,  and  2  feet  3  inches  wide  ? 

By  Duodecimal*.  By  Decimals. 

F.    I.  F.    I. 

16    9  16    9=16575feet. 

23  23=  2/25 


S3    6                      «.  8375 

423  3350 

S350 

37    8    S  F.    /.    * 

Oft  *fns.  37,68751-37    8    9 


230  APPEXDIX. 

TO  MEASURE  LOADS  OF  WOOD. 

Kr 

MuNipljrthe  length  by  the  breadth,  and  the  product  by 
the  depth  «r  height.  ,vhtc!i  v/ill  j^'m:  the.  content  in  solid 
i'eet;  01  wiiicu  '.,ilt  a  ccrtl,  and  128  a  cord. 

VMPI.I:. 

Ilinv  many  solid  feet  are  contained  in  ;i  load  of  wood, 
7  fee;  (  :g,  4  feet  2  inches  wide,  and  2  feet  3 

7ft.  <jin.^7,~y  and  4  ft.  2  I'M.  —  4.U>r  nnil  2ft.  f>  tH=a 
C,£5  :  (hen,  r,5x4,l67=51.io2.>x0025==rO,318l25  so/ui 


IJnt  '-tads  <;f  wood  are  commonly  estimated  by  the  footj 
allowing;  tlie  load  fo  be  8  feet  long,  4  fe^t  wide,  and  (hea 
£  feet  high  wiil  make  half  a  cord,  which  is  called  4  feet  of 
wood  ;  !);i(-  if  the  breadth  of  the  load  be  loss  titan  4  fect| 
its  hi  -t  l>e  increased  SD  as  to  make  l>alf  a  cordj 

\vl;ic;i  i.     till   called  4  feet  of  wood. 

By  r  ihc  bre-ultli  and  hei^hthof  the  load,  tin 

content  may  be  found  by  the  following 
"RULE. 

Multiply  the  broa:l!!i  by  thii  hei^l'it.  and  half  the  pro- 
duct will  be  t'te  content  in  feet  and  inches. 
EX  A  51  PL.  B. 

Required  the  content  i-i  a  load  of  wo'id  u-hicli  is  .3  feet 
9  inches  ^\ide  and  ^  ioet  ()  inciios  high. 
By  Duodecimals,    Ly 
F.  in.  F. 


F.  In. 

=4     8±,  or  liti[f  a  cyr..' 

8  \  iuchr*  vfer. 
tirf - 

i-c!  '  •.  h't  the  follpwlng Tnlile  M 

of  nny  1AA.)  uf  w  ">l.  by  ii^|«odi<'i' •  -,taU 

.  iirnctict!  ;  ^'.'cti  wal!  lie  <<«inJ  \ 


Al'PKXDIX. 


231 


Jt  T.1BLE  of  lirfartth,  Tf eight,  and  Content. 


Breadth. 

Height  > 

/flC/JM. 

'ft.  in. 

1 

o 

S 

4 

3 

4 

5 

6  |7|H 

9  flOfll 

2    6 

Ju 

3U 

4o 

1 

4 

5 

6 

7 

9 

10 

11 

12 

14 

7 

3  1 

47 

1 

3 

4 

5 

6 

8 

9 

10 

12 

13 

14 

8 

u. 

48 

64 

1 

3 

4 

£j 

7 

8 

9 

11 

12 

13 

15 

9 

i; 

33 

49 

1 

314 

6 

7 

8 

9 

li 

12 

14 

10 

17 

54 

51 

2 

3 

4 

6 

7 

9 

10 

LI 

45  14 

16 

11 

IS 

3Ji 

ru 

2 

3 

4 

6 

7 

9 

10 

12 

13  15 

16 

5     0 

18 

72 

2 

5 

6 

8 

9 

11 

12 

17. 

1 

19 

37 

.76 

74 

2 

3 

5 

6 

8 

9 

il 

141'.: 

17 

2 

19 

70 

0 

S 

5 

6 

8 

10 

1  ! 

13 

17 

3 

19 

78 

o 

3 

5 

7 

8 

V 

11 

13 

15 

18 

4 

20 

40 

80 

a 

3 

g 

y 

8 

10 

12|Io 

15 

17(18 

5 

£1 

41 

62 

8:! 

3 

5 

•» 

i 

10 

17|19 

(5 

63 

84 

£ 

4- 

7 

9 

ir2 

7 

43 

64 

o 

4 

5 

7 

9 

11 

••  0 

16 

20 

L         8 

22 

66 

7 

9 

11 

15 

17 

18 

eo 

'        9 

23 

68 

4 

7 

9 

15 

15 

17 

19 

21 

10 

23 

46 

69 

^ 

4 

6 

7 

9 

1!) 

21 

11 

23 

70 

(> 

3 

10 

16  (b 

20 

22 

4     0 

24 

4S 

72 

96 

6 

8 

10 

12 

29 

TO     '  GOING    TA< 

First  measure  thr,  u;e,ul..:i  and  he^a.  of  yuur  load  to  the 
nearest  average  inci.  -.1  the  bre:-  hand 

column  of  the  table  ;  t'.ion  movett  tli'i  right  on  the  s^mj  line 
till  you  come  under  the  height  in  fret,  a. id  yflu  ui:i  '.nvo  the 
content  in  inche j,  ru.  •,  IM  i,I.'.  con- 

tent of  the  inriies  on  tli  ••  <iivi«le  the  sun  hj  1^,  and 

you  will  ii.ivo  tlift  true  COM'.-'','!  uf  I::  u-s. 

NOTI:.— The  contents  a;)^.v,;ii:^  the  inches  being  always 
small,  may  be  added  by  inspection. 

EXAMPLES. 

1.  Admit  a  loa^  of  WQH»!  is  3  .ect  t  inches  v>ide,  and  2  feet 
10  inches  hi^h  ;  rf-r;uired  t'  ?  r'.ntent  — 

Thus,  against^  fl.  4  inche?.  aH  is'.i.li-  •  )  inch- 

es ;  and  under  10  inches  .-  ;04~ 

17=57  true  content  in  inches,  which  divide  by  li  givvs  4  fert 
9  inches,  the  answer. 

£.  The"  breadth  being  3  feet,  and  height  2  feet  S  inches; 
Kequired  the  content. — 

Thus,  with  brsaJth   3  feet  0  inches,  and  under  2  feet 


AF9EKDIX. 


Atop,  stands  36  inches ;  and  under  8  inches,  staidif  It 
inches :  now  56  and  12,  make  48,  the  answer  in  inches  $ 
and  48-j-12=4  feet,  or  jwst  half  a  cord. 

3.  Admit  the  breadth  to  be  3  feet  1 1  inches,  and  heigbt 
3  t'-et  9  inches;  required  the  content. 

I'udei-  3  feet  at  top, stands  70 ;  andundtr  9  inches, is 
.'0  and  18,  make  88-i-12=7  feet  4  inches,  or  7  fit.  1 
•-jr.  ;?  inches,  the  answer. 


TABLE  I. 

the  amount  of  £  1 .  or  gl,  at  5  and  6  per  cent,  per 
annum,  Compound  Interest,  for  20  years. 


FT* 

5  per  cent.\6per  cent.\I'rs.  15  per  cttnt.\6  per  cent. 

1 

1,05000 

1,06000 

11 

1,71034 

1,89829 

2 

1,10550 

1,12360 

12 

1,79585 

2,01219 

3 

1,15762 

1,19101 

13 

1,88565 

2,13292 

4 

1,31550 

1,26247 

14 

1,97903 

2,26090 

5 

1  ^7G'28 

1,33822 

15 

2,07893 

2,39655 

6 

1  ,34009 

1,41851 

If) 

2,  is 

;727  i 

7 

1,40710 

1,50363 

17. 

2,29 

1277 

8 

1,47745 

1,59384 

18 

661 

9 

1,55132 

1,68947 

19 

>95 

,  10 

1,62889 

1,70084 

20 

2,G. 

11 

5 
2 
17 
8 
4 
1 

8 

4 

Gl 

15    L 

10JJ 

16    1 

8  r 

I7f 

Sta;v 

(,: 

..fard 
Silver. 

VII.    The  weights  qf  the  coins  of  the  Unite  J 


Baffles, 

Half-Eagles, 

Quarter-Eagles, 

Dollars, 

Ha!M)»Ilais, 

Quarter-Dollars, 

Dimes, 

Half-Din 

Cents 

Half-Cents, 

Tie  standard  fur  gold  cc  n  is  II  parts  pure  gold,  ajid  one  part  al- 
ley.— the  (Hoy  to  x  onsivt  ol'  iriver  and  copper.  The  standard  for 
*iiuT  coin  ii  I486  parU  fine  to  179  parts  alloy — the  alloy  to  be  whol- 
ly oopp«r. 


V.D1X. 


•ANNUITIES. 


TABLE  II. 

TABLE  111. 

;jjg  th?.  ammwt  of 

'he  present 

jfl  annuity,  fur' 

Vjorth  of  £1  annuity, 

".  I  pars  or  «i 

to  con,in;ts  fin-  5ll 

at  5  and  6  per 

yci?*,  cd  5  and  6  per 

compound  in??mst. 

I'rs.    5 

5 

1 

1,OG< 

1,OC 

J581 

2 

1.85 

3,15 

oV  2; 

rsois 

4 

L0125 

4616 

3,545950 

.HOG 

5 

4,2i23G4 

6 

4,917524 

f 

8,1- 

.  :!3838 

6278 

5,5' 

s 

.  .'Jill!) 

6,463213 

6,20 

9 

11,026564 

11,491516 

7,10 

6,80  1  692 

12,577892 

15,180770 

7,721735 

7,360037 

11 

14,206787 

14,9716-13 

8.506414 

T  Q  <•  - 
4  «**Q\-'D  i  »/ 

12 

15,917126 

16.869942 

.  -J3252 

B,S83844 

IS 

17,712982 

18,8821  .IS 

9,593575 

8,852683 

14 

19,598632 

21,0150JC 

9,898641 

9,2: 

15 

21,578564 

25,27 

10,379658 

9.712249 

16 

23,657492 

25,672526 

10,837769 

10,105895,! 

17 

25,840366 

28.212580 

ll,27406u 

10,47? 

18 

28,152585 

30,905653 

11,689587 

1  0,827603  1 

19 

30,539004 

33,75990-2 

12,085321 

11,158116 

20 

35.065954 

36,785592 

12,462210 

11,46! 

21 

35,719252 

59,992727 

!  12,821  155 

i  1,764077 

22 

38,505214 

45,592291 

15,165005 

12,041582 

23 

41,430475 

46,995823 

!  15,488574 

12,30 

l-4 

44,501999 

50.8I557.S 

il3.79S642 

12.550557 

25 

47,727099 

54,8645  12J 

14,09394-iji2,7So35ti 

26 

51,115454 

59,1563821 

14,375185 

13,003166 

27 

54,6691  £6 

63,705765 

14,645054 

13,210534 

28 

58,402583 

68,528112 

M.M-  i  .: 

1.3,406164 

29 

62,322712 

73,659798 

15,K 

j  5.590721 

SO 

66,438847 

79,058186 

15,57?451 

!.i.r64851 

*31 

70,76079084,8016771  15,592810 

13.929026 

20' 


254 


APV1VBIX. 

TABLES/ 


1  HE  three  following  Tables  are  calculated  agreeable 
to  an  Act  of  Congress  passed  in  November,  1792,  making 
foreign  Gold  and  Silver  Coins  a  legal  tender  for  the  pay- 
ment of  all  debts  and  demands,  at  the  several  and  respec- 
tive rates  following,  viz.  The  Gold  Coins  of  Great-Bri- 
tain and  Portugal,  of  their  present  standard,  at  the  rate  of 
100  cents  for  every  2^  grains  of  the  actual  weight  there- 
of.— Those  of  France  and  Spain  27|  grains  of  the  actual 
weight  thereof. — Spanish  milled  Dollars  weighing  17 
pwt.  7  gr.  equal  to  100  cen-ts,  and  in  proportion  for  the 
parts  of  a  dollar. — Crowns  of  France,  weighing  18  pwt. 
17  gr.  equal  to  110  cents,  and  in  proportion  for  the  parts 
of  a  Crown. — They  have  enacted,  that  every  cent  shall 
contain  208  grains  of  copper,  and  every  half-cent  104 
grains. 

TABLE  IV. 


Weights  of  srreral  pieces  of  English,  Portuguese,  and 
French  Gold  Coins. 


Pwt. 

Gr. 

Dols.    CtS.  Jlf; 

18 

16       0      0 

9 

8 

tintlish  Guinea,  .... 
Half,        ditto,  

5 

a 

6 
15 

4      66f 
2      33$ 

French  Guinea,   .... 
H.\lf,       ditto,    ..... 

5 

0 

6 
15 

4      59      R 

2     29      9 

4  Pistoles,   

16 

12 

14      45      2 

2  Pistole*,    

8 

6 

7     22     6 

4  Pistole,  w  . 

4 

8 

S      61      S 

Moitlore,   •  . 

6 

22 

6      14      8 

APPENDIX.  > 


;js 

55   „• 
3g 

£§2 


2: 


coo 
—  o 


.. 


O  O  --; 

j 
—  <?»«-*  u-  • 

«c  G»  a.  "5  e-t  oo  »o  «.  co  «f  >-«  to  •*  c  r~  (•": 

—  •*  oo  c»  o  •?>  tr>  -. 
o  *o  *o  ?c  o  o  c^  <^ 

—  etw^oscr-esc;  o  —  or:-*O'jr-o(?— .  s- 

„  _  _  ^,  c , 

.^— — ^_  ( 
O  -*  G»  f5  •*  o  cs  c~  C3  ci  O»  o  —  (?)  1-3  -?  o  c:  :o       t~  o  « 

—  -i  —  — i          —  C7  O  ' 

iS   r«   i-   — 

ct      o  T  —  ct  •«  sr 

J"-  O  ~*  ^  I  ^ 

—  —  —  —  t)  ,. 


2:-.  6 


Mil.  T3BLE  of  Cants,  ansa-cnnc;  to  tlie  Currencies 

of  tkt>  United  ftt,''!rs.  K-ith  Sterling,  <$'c. 
No  rv.. — The  fi<;;:;c>3  o-i  1:>e  right  hand  of  the  space, 
show  the  parts  of  :i  cent,  or  mills,  &c. 


6a.  r&lKs.  f(.> 

is.  /» 

4s.Gd. 

4s.  10}//. 

the 

!<>  fin- 

the 

!o  ilt- 

to  the. 

• 

Doll. 

Dull.  \  JJolL 

Dollar. 

r. 

rente.  (•<?)•'.<>•. 

cents. 

cent*. 

< 

\ 

1   5 

1  0 

I   I 

J   7 

1  C 

1  S 

1     7 

& 

C  7 

2  0 

2  ii 

3  5 

5  7 

3     4 

4   1 

5  1 

3  5 

5  S 

5 

6  5 

5     1 

4 

5  -5 

4   1 

4  -1 

7  1 

7  4 

6     8 

5 

6  9 

5  2 

5  5 

8  9 

9  2 

8     5 

•C 

8  3 

(i  ii 

6  G 

10  7 

10 

11   1 

10     2 

7 

9  7 

7  2 

7  T 

12  5 

11  C 

13  9 

11     9 

8 

11  1 

8  3 

8  8 

1-1  2 

13  S 

14  8 

15     6 

9 

12  5 

10 

16 

15 

16  r 

15     3 

10 

13  8 

10  4 

1!    ! 

17  ¥ 

16  C 

18  5 

17 

11 

15  2 

11  4 

12  2 

19  6 

18  S 

20  5 

18 

S. 

1 

16  G 

1°.  5 

13  5 

21  4 

20 

22  2 

20 

2 

33  5 

25 

i,6  f 

42  8 

40 

44  4 

41 

- 

37  5 

40 

64  2 

60 

66  6 

61     5 

4 

66  6 

50 

53  3 

85  7 

80 

88  8 

82 

5 

83  2 

li'2  5 

66  6 

107  1 

100 

111   1 

102     5 

ti 

100 

75 

128  5 

120 

133  5 

125 

<j 

U6  6 

ic  r, 

95  3 

150 

140 

155  5 

145     5 

8 

100 

171  4 

160 

177  7 

164     1 

9 

150 

112,5 

1'20 

192  8 

180 

200 

184     6 

10 

166  6 

125 

214  2 

200 

222  2 

205     1 

1! 

183  5 

137  5 

146  6 

255  7 

220 

244  4 

225     6 

1:2 

eoo 

I  GO 

257  1 

240 

266  6 

246     1 

13 

216  fiji(i2  f< 

173  3 

27K  5 

260 

288  8 

266     6 

1-* 

300 

311  1 

1 

IS 

•250 

1S7  5 

200 

521  4 

500 

335  G 

307     <i 

;        1    • 

200 

213  5 

342  8 

320 

31J8     0 

17 

212  5 

226  6 

364  2 

340 

577  7 

548     7 

19 

300 

225 

240 

385  6 

560 

400 

569     2 

!     19 

,)  I  ii  6 

257  5 

255  5 

407  1 

580 

4  22  2 

589     7 

|     20 

353  5 

250 

266  6|428  5 

400 

444  4 

410     2 

APPEMDIX. 


237 


TABLE  ix. 

Showing  the  value  of  Federal  Jloney  in  other  Currencies. 


'Federal 

Money. 

New-Eng- 
land,   Vir- 
ginia, and 
Kentucky 
currency. 

.Vew-rork 
snd'j\T(irtii~ 
Curvlina 
currency. 

t/V.  Jersey, 
Pennsyfoa- 

nix*  i)da- 
«•«;•.•',  and 
Maryland 
currency. 

Soi*it--Car- 
(.'.';:&,  «;irf 

•£•/« 

currency. 

Cents. 

s.     d. 

s.     d. 

s.      d. 

s.      d. 

1 

0      Of 

0      1 

0       1 

0      04 

2 

0      I1 

0      2 

0       1} 

0       1 

S 

0      24 

0      3 

0       2$ 

0        Ij 

4 

0      5 

0      3^ 

0       34 

0       2-i 

5 

0      Si 

o    42 

o     44 

0 

6 

0     4$ 

0      5} 

0       54 

0       34 

7 

0     5 

0     6^ 

0      64 

0       4 

8 

0      52 

o    71 

0      74 

0      44 

9 

0      64 

0      82 

0      8 

0       5 

10 

0     74 

0     94 

0      9 

0       5J 

11 

0     8 

0   104 

0     10 

0       64 

12 

0     8| 

o  114 

0     102 

0       6j 

IS 

0     94 

1     04 

0     llj 

0       74 

14 

0    10 

1     14 

1       04 

0       7| 

15 

0    10* 

1     24 

1      14 

0      84    • 

16 

0    1U 

1     34 

1      24 

0       9 

17 

1     04 

1     44 

1      34 

0       94 

18 

1      1 

1      51 

1      44 

0     10 

19 

1      1* 

1     64 

1      54 

0     102 

20 

1      2} 

1     74 

1       6 

0     114 

so 

1      9i 

2     4| 

2      3 

1       4J     . 

.   40 

2     42 

s    24 

S       0 

1     104 

50 

3      0 

4     0 

3       9 

2       4 

60 

S     74 

4     9* 

4      6 

2       9} 

t     70 

4     24 

5     74 

5       3 

3       54 

80 

4     94 

6     4* 

6      0 

S       82 

90 

5     42 

7     24 

6      9 

4       3i 

I  100 

6     0 

8     0 

7      6 

4       8 

A   FEW    UsEFLI.   fOKMS    IN  ')  R  '-NSAC  fJN-    DC  SI  NESS. 

AN  OBLIGATORY  BOND. 
KNOW  all  i.    n  i-v  i:ic-  .     .  I),  of 

(»f  a'll  .ildtO 

i  i  of  •  paid 

iM^aii-1  ;idi  iinisir;it 

tl,  v>.  \\i-;l  ai.il  :   ..:;   to  l.t-  r  ..rie  ;:nd  -.lone, 

1  br  T-    MU!  admtnistratMS, 

•d   v.-.tli   my  baud,  aud 
i.     1-utod  at  tliis  day 

«f 

T//e  conchii  i  "f  t'l't  •  obtigato  .  h.  That  if  the 

above  bound*  a  (.'.'I).  &c.   i  tht  c->}idiiio)i.] 

Then  thi-  in  t ••  be  v.,u!  ami  of  T'.or.e  effect;  otlier- 

Mi-,0  :  .d  \  i.fuc. 

lied  and  delivered 7 
in  the  pretence  of  5 

A  BILL  OF  SALE. 

KNOW  all  men  by  \  .  iliat  I,B.  A.  of 

iid  in  ci<:.  hand  ]>aid  bj 

1).  C.   tit'  tlic   ro;  :ipt  \vlifc;vof  I   an  iicreby  ac- 

knr. .  .   e<!,  .-;>ld  a.-d  d.  iivered,  aud,  by 

^•nts.  ii<.  bargain, :-  i'.i  er  unto  the  said 

O  v  •      ;  >:Wrf.]  To  HA\E  and  to 

HOLD  the  aforetaidbarraiaed  prei ..     -.  ur  ,tl  I). 

1\  bis  execut-x--.  -   -c  '   Ufl  .'i  ,   forever. 

And  I,  the  ;-aid   1$.  A.  f.r  inytell.  my  exert  v»rs  and  ad- 

iliii:'r  -'iali  :i!i<l  \\..i  v/ai   at:t  BJ    1  «U-iV:.a  tUc  Same 

nst  ail  persons, 'untfl       -  '        >exec  ''us, ad- 

'•i     4     :  *.     In  -uitnest 

>vfct-  \c  her^"nto  set  :..)  hand  aud  s<^al,  this 

;       d.  1814. 

In  jirestncc  t>f 

A  BHOKT  WIZX. 

I,  R.  A.  of,  &c.  d»n)..h.e  rncL  «       i.t^ismv  last  will 
ard  testamcat,  in  manner  and  form  follcnviiig,  viz.  I  giv» 


APPENDIX.  £G9 

and  bequeath  to  my  dear  '>:•<. thcr,  R.  A.  the,  sum  of  ten 
pounds,  to  buy  him  mourn' ng.  1  give  and  bequeath  to 
ray  son,  J.  A.  the  sum  of  two  hundred  pounds.  I  jrive 
and  bequeath  to  my  daughter.  K.  II.  the  sum  of  otic  hun- 
dred pounds;  and  t.»  in  •  r  A.  V.  the  iik<>  s;.;n  of 
one  hundred  ;ioun<;  ,.  AM  the  rest  and  residue  of  inv 
estate,  goods  and  chattels.  I  givo;  and  bequeath  to  u;y 
dear  beloved  wife,  E.  R.  whom  I  nominate,  cutibtitute 
and  appoint  sole  executrix  of  this  mv  last  v  ill  and  tes- 
tament, hereby  revoking  all  other  and*  former  wills  by  me 
at  any  time  heretoiu.  e  made.  In  -witness  whereof,  1  have 
hereunto  iet  my  hand  and  seal,  the  day  of 

in  flie  year  of  our  Lord 

Signed,  sealed,  published  and  declared  by  the  said 
testator,  B.  A.  as  and  lor  uis  last  will  and  testament,  in 
the  presence  oi  us  who  have  subscribed  our  names  as  wit- 
mcbsesi  thereto,  in  the  presence  of  the  said  testator. 

R.  A, 
S.  D. 
L.  T. 

NOTE. — Th«  testator  after  taking  off  his  seal,  must  In 
presence  of  the  witnesses  pronounce  these  words.  "  I  pub- 
lish and  declare  this  to  be  my  last  will  and  testament." 
Where  real  estate  is  devised,  three  v  i  ,!ie---"s  are  abso- 
Intel  v  necessary,  who  must  sign  it  in  the  presence  of  the 
testator.  % 

A  LEASE  OF  A  HOUSE. 

KNOW  all  men' by  these  presents,  ujat  J,  A.  B.  ot 
in  for  «ud  in  consideration  oi'  the  sum  of 

received  to  my  full  satisfaction  of  P.  V.  of 
this  day  of  in  the  j ear  of  our  Lord,        have 

demised  and  to  farm  let,  am! "do  by  these  pie.  ert.=  ,  de- 
mise and  to  farm  let.  unto  the  saul  P.  V.  his  heirs,  esecu- 
tors, administrators  and asstigns,  one  certain  pi'-ceof  'and, 
lying  and  be'nuj;  -i;nn:-d  in  -aid 

[Here  describe  the  boundaries]  with  a  duelling-huii  .Q 
*horeon  standing,  for  the  term  of  one  yea;  frou  t!  ,3  (lai.e. 
To  UAVB  ;ind  to  H.n.n  r«»  him  t:  c  said  P.  V.  his  hens?, 
ivecutors,  linl-u?  -•  •!.•;.  ,  ,s  l«r  «?]•!  r^vm,  tor 


240 

lim  the  said  P.  V.  to  use  and  occup1-   at  to  i.iin  shall  seem  meet 

u:i  \  proper.    And  the  said  A.  B.  doth  FURTHER  COVENANT  with 

\i  P.  that  he  hath  good  ri^ht  to  let  and  demise,  the  said 

l"ttcn  and  demised  prjmise.s  ift  manner  aforesaid,  aud  that  JMJ 

1  A.  during:  the  said  time  will  suffer  the  s^iid  P.  quietly 

f.i  H.\vr  -md  to  HOLD,  use,  occupy  and  enjoy  said  demised  pre- 

..it  ^ui  1  P.  shall  haw,  hold,  use,  occupy,  possess 

and  <  nioy  the  same,  free  ami  clear  of  nil  incumnmnces.  claims, 

In  witness  whereof,  I  the  said 
.\.   Ji.  i.uve  hereunto  set  my  hand  ami  seal  this 

SigntJ,  stnlfil  and  delivered  )  4    R 

$ 


In  presence  of 


A  NOTE  PAYABLE  AT  A  BANK. 

[$500,  60]  HARTFORD,  May  30,  1815. 

FOR  value  rrcc-irod,  I  promise  to  pay  to  John  Merchant^ 
«r  order,  Fivo  Hundred  Dollars  and  Sixty  Outs  at  HaxtfoTd 
Dank,  iu  sixty  davs  from  the  ua'.f. 

WILLPAJI  DISCOUNT. 


AN  INLAND  BILL  OF  EXCHANGE. 
[483,  34]  BOSTON,  June  1,  iai.'. 

i;NTY   days  7tftcr  date,   ploase   to  pay  to   Thuma* 
iii   or  order,    r.ijlity-Tliree   Dollars   and    Thirty-  Fonr 
:  ud  place  it  to  my  account,  as  per  advice  from  your 
.  >  sen-ant,  SIMON   PI  fk 

Mr.   T.    W   Merchant,  ) 

Wat-York.  • 


A  COMMON  NOTE  OF  HANI). 

,;irrh  t;,  i;;21. 
•  to  pay  to  .lulu;  Murraj, 
in  four  cionths  from  th 
JO; 


A  (  T,R. 

! 
•  'l.nrle*  <"; 

• 
TV.  rut  . 

' 


THE 


PRACTICAL  ACCOUNTANT 


.KKST  METHOD  01" 


INSTRUCTION  OF  YOUTH. 


AS  A  COMPANION  TO  DABOLL'8 

ARITHMETIC. 

IJY  S  \MUEL  GREEX. 

%  * 

rUBUSIIED  BY  SAMUEL  GREEN, 
XEW-LOffDON. 


INTRODUCTION. 


.       ,  :ul  fomalo,  - 

T'lim-fic,  ospcrinily  in  the 
r<(  .iliil  rn!.-s  .->f  AMI'*'--  >!i.  MI!,  tract  ion,  Multiplication,  n:- 1  Di- 

-!if)nl«l  be  instni  •      noeof  BooA-  A 

'  fiuj fitter  of 

.   •  •ir-nly 

i--i  '  by  t}«?  men-' 
• 

i[>!o  ed    in  :.« -quiring  a 
rat. 

•'t'  ill"'1,  \vbo 
' 

;UlJ    COTTC'Of 

Le  ;-•:'  ]>:i]-pr 

•  ml'-d  aiV  ,  -y-tom.    Ii> 

'  .)!.,  :n-i:  duly  to  *        >      •    .       .1    . 

all  tin-"  "  which 

- 

on  ;iny  ron- 
i  shoultl 

•i         ..  ,  liia'i ;    thfi  rn<»-> 

' 

•    . 
n  to  whom 

rsJ  :    he" 

. 

1 

• 


• 


. 
i 

Entered. 

1 
» 

Fnleral. 
1 

Entered. 
1 

Entuml. 
i 

Entered. 
1 

Entered. 
1 

Entered. 

1 

Entered. 

Cr. 
date,         ..:... 

r 

5 

18 

1 

11 
U 

100 

'> 

:i 

3 

2 

,-iO 

.30 

-.!! 

G4 
:10 

Sarmicl^:.                  .         .         .Dr. 

• 

:i',v\rk 

this  ,h'.v,         ..... 

Ur. 

To  n                               is  out  of  the  stuiij  ot'j 

All! 

An;'.  of)  v  Uillir                .          .            Cr. 
H\  iiiv  urv!(v  in  !...  .,-r  ol  J<                      i^s, 
i  ~                _ 

1).-. 
completed  and  i 
tlii=   lay  <"i  l;i;>  (-•  i;  •                   sj  called, 
4000  foet  at  2J  cents  per  foot,      . 

-i  }.> 

Et>.                            .         .         .        Cr. 

liy  bis  team  ut  sund'-y  tiaiv 
nure  on  my  lan.i.         .... 

or 

Thornas  Grosvcnor,         .         .         Dr. 
To  48  \vindo-.v  sishcs  delivered  Pt  hi1*  C.lovrr 
Farm,  ?o  called,  at  4*1  00.   .       .     ^ 
1  |  ...eaoi'ijlass  by  in. 
Job'i,  :-.t  li  cents,          .         .          7,50 
10  daj  s'  *vork  ui'imself  finishing  front 
room,  at  jgl  ,~j  a  day,     .         .       1J..OO; 
7i  »Io.  of  WiMiam,  my  hired  man.  i 
laying'  the  kitchen  floor  i-iul  liauvj-  >  6,30 
iug-  doors,  at  C4  cci't^  a  d-i  v,            )  

An*                              .         .         .     Cr. 
By  2                                        's.  pfM-pv.ll.  0,7  j 
ds.  of  lii.li:i  Cotton,  a!  I.';/,  cents^  0,74 
2  flannel                      cph  li^tnip',     ^,li^ 

Dr. 
Toii  • 

Thtre  put  tin,  name  if  iht  wrier  ij~  the 


««/»,:'_• 


FORM  OF  A  DAY  BOOK. 


Albany,  February  12, 


Entered. 

1 

Entered. 
1 


Entered. 

1 


Eutered. 
1 

Entered. 

1 

EnJowd. 

1 


Edward  Jones,         .         .         .       Cr. 
By  4  months'  hire  of  his  son  William  at  $10 
a  month, 

-24 


Edward  Jones, 
To  my  draft  on  Thomas  Grosvenor, 


Thomas  Grosrenor,        .        .        c'r. 

By  my  order  in  favor  of  Joseph  Hastings, 


Joseph  Hastings, 
To  my  order,  on  T.  Grobvenor, 


Dr. 


Thomas  Grosvenor,        .         .        Dr. 
To  3  days'  tvork  of  myself  on  your  fence 
at  gl',25  per  day,         .         .         .      3,75 

Jo.  my  man  Wm.  on  your  stible 
and  finishing  off  kitchen,  at  84  cts.     2.52 
2  pr.  brown  yarn  stocking?*,  at  42  cts.  0,34 

13 ~ 


Dr. 


Thomas  GrosvcDor, 
By  my  draft  in  favor  of  E.  Jones, 

23 


Cr. 


Thomas  Grosvenor,         .         .        Dr. 
To  part  of  u  dav's  work  of  my  son  John 
.-nan  William,  on  bis  barn, 


Anthony  T.iUinpj,  .         .         .         Cr. 
For  :.je  following  urti^les, 
1 4  Ihs.  n  •    «rado  sugar  at  $12  pr  cwt  1 ,50 
dish,        ....       0,23 

C  pi;ue,, 0,3° 

ucers,      .         .         .       0,20 

1  pint  French  Brandy,   .        .        .0,17 

i  'juart  Cherry  Bounce,      .         .       0,33 

•31>e,    ....  0,18 

bles, 0,04 

1  piir  Scissors,        .         .         .         .0,17 
1  qnire  paoer,      .         .         .         .       0, 
\Vaf«w,  4  \  ink,  6  ;  1  botUe,  8  ;       .  0. 1 8 


Jtatored.       I'et  ,-r  Daboll,  ....         Dr. 

1  JTu  a  cotton  Covtrit>t  deLvcrcd  Sarah  Brad- 
1    ford,  by  your  written  order,  dated  14.  Jan. 


FORM  OP  A  DAY  COOK. 


L, 


Entered,  i     Tliunius  Crtisvcuor, 


Cr. 


i 


Entered. 
1 


Entered. 
1 


Entered. 
1 

Entered. 
i 


IkulereJ. 


. 


Dr. 


ii  v  BilHi'L---, 
'  .   pw  1  of  Ciller,       .         .        $1;17 

1  barrel  containing  the  r.'iiie  (from 
TiKHir.ii  Grosvenor.)       .         .         .  0,f>f; 


Tiiomas  G:  •. 


Cr, 


Uv  i  biin-el  •_  iider  aold  aa<I  deliv- 

ered to  Authors 


10 


•dj  IL 


Dr. 


To  cash  per  his  order  to  George  .Gilbert, 


}\ter  Daboll,  Cr. 

By  amount  of  his  shoe  i.r.count,     .       vjl,  l(J 
Yarn  received  from  him  for  the  bal- 
ance of  his  account,       .          .          .1 .03 


Eutered. 


Eatoro*]. 

2 


Sarimtl  Gire.ii,        .         .         .         Cr. 
By  amount  due  for  12  mouths  New- 
London  Gazette,       .         .          .       §2,00 
4  Spelling  bookb  at  20  cts.  for  chil- 

dren,        .....    0,1>0 
1  Daboll's  Aritliioctic,  for  my  son 

Samuel,       ,  0,  1'-' 

£  Blai.h  Writiog  books  at  12J  ccnt«,   0,'25 
)  quire  of  Letter  Paper,       .         .     O..M 


.N.        :..\;it)ie,          .         .         .       Cr. 

e  of  viii  iatc  endorsed  by  Ephraim 
Dodg'o,  ..t  C  niontl  ^,  lor  a  yoke  of  Oxen 
bought  of  Lujaiol  Mason,  at  Lebanon, 


Joiiuthan  Curtis,       .         .         .      Dr. 
To  an  old  bay  horse,         .         .         $93,OOJ 

a  four  wheeled  w;!£gon,  and  half 

worn  harness,        .         .         .       42,00' 


Entered.       Samuel  G  reru, 
2   To  cash  in  f 


3*81 


FORM  OF  A  DAY  BOOK. 

""Albany,  AprU  6,  1822. 


Anthony  billing,          .         .          .   Dr. 

:  $11,25,     .       .  £22,50 
Amount  of  order  dated  March  2Cth,  i 

iniuvow  of  I  aiuiv  White.  >0,o4 

"uir  yarn  .stockings,         j 
Hire  v.f  »i!y  waggon  and  horse  to  i 
brii;.  :Vo:jn  Provi-  >3,00 

ut  Uiis  jnont]>,       .        N  — 


Thomas  Grcivenor.       .         .       .   C'r. 

1    |Ry  iiis  order  oa  Tln'odorc  15arrcJ3,  ? 
•a  \>r  Go  d< 


ItntcrcJ. 


26*04 


'jj  08 


.      Dr. 
To  1 

ills.  alSOceuts.       .       .  $60,OOJ| 

..mi  s;uj  Barrel  1  fur 
balance  due  0:1  Thomas  (Irosvc- 
nor's  order,  .  .  .  .  18,0fl 


—18- 


2    By  « coat 

lutercil. 


Entfrcd. 
1 


Jou;i(li:»a  ('urtis,        .          .          .       Cr. 
7i,  p:inta)oon^  g">,00, 


Thomas  (» rosvcrtor, 


Dr. 


To  mending1  vour  cart  by  my  man  Wil- 
liam, '.  .  .  . 

Paid  Hunt,  for  blacksmith's  work  on 
your  cart.  ....  0 

St-itin;r  '.;  pfllc*  i>f  cla-.s,  and  finding 
glass, O.GG 

Oj H 


Dr. 


. 


Antliony 
Bv  pa-  . 


1 


•  • 

.     7,a^ 

ilas- 


Oil,  per  t/. " 


>'    •  ... 

• 


i 

. 


or  A  BAY  BOOK. 


Albany,  Muv3,  182-.?. 


Kntercd. 
._> 

Entered. 
] 

Kntorcth 

1 

Etitoral. 
1 

Entered. 
2 

Entered. 
1 

Kntercd. 
1 

Knterod. 

'•I 

Theodora  Jiarrcil,  New-.Lon  Jou,     Dr. 

To  1.)  die?  •••  ,  '»>•'!  Ibs.  at  5  cent..,                • 
L'17  Ibs.  of  butter,  at  13  2-3  ccuts,u34,GO 
24  Ibs.  of  h<)ii'\v,  at  12J  cents, 

i 

52 

1 

* 

43 
SI 

52 

54 
48 

c. 

-10 
25 

60 
50 

40 

00 

no 
oe 

Joseph  Hustings,     ,         .         .         Dr. 
:\'  1   i\iir  t-liocs,  :J9th  Apnl,  frofft  Anthony 

i    -  -     12 

Anthony  B&iags,    .        .        .      Dr. 

'l.'o  !M  bushels  of  seed  potatoes,  at  33  l-'3 
<\-uts,         ...           . 
f,  pair  mittens  at  20  cents,       .       .1,60 
(.  i'-h,         14,00 

Joseph  1  .                      ...    Cr. 
iJy  4i  months  wages  at  7  dollars, 
"0 

•   Theod  >ri>  Karrell,           ,          ,         Cr. 
By  cash  in  full  uf  all  demands,       . 

Thonv.'s  Grosvenor,         .          .         Cr. 
by  his  acceptance  oi'  my  order  in  favor  of 
Anthony  JJillincrs,         .... 

Anthony  Billiu.'^,     .         .         .       Dr. 
To  amount  of  my  order  on  Thomas  Orosvc- 
nor,         ...... 

Si  nt   "  1 

NoN-s  pjn-abk-,         .         .         .        Dr. 
To  cash  paid  for  my  note,  to  D.  Mason, 

The  foregoing  exntnple  of  a  Day  Book,  may  suffice  "to  give  a  good  ideaoi' 
*}]f  w  iv  iii  which  it  is  proper  to  make  iho  original  entries  of  all  dobt  and 
cTcdit  articles.  Another  small  book  should  next  be  prepared,  according  t» 
the  followin};-  lonn,  termed  the  book  of  Accounts,  or  Legor.  Into  this  book 
must  bt:  posted  the  whole  contents  of  tin  D;iy  Book;  care  being  taken  that 
» vi TV  article  be  carrii^d  to  its  com-,pon«linc;  title  ;  the  debt  anioiini*  'o  b« 
entered  in  the  left,  aru  the  credit  in  the  right  hand  page1.  Thus,  should  it 
st  nny  time  be  required  to  Un«*w  tlie  staff  c>f  an  nrivint.  i*  "'ill  only  l>e  n»- 
o  -*-iiy  •  •  olamus,  and  <o  M  -..i;»llir  amount 

ii'i'in  ilie  greater,  the  remainder  will  be  the  balance. 

When  an  article  is  posted  from  the  Daj  Book  into  (hf  Lrprcr,  it  will  b* 

prop«r,  opposite  the  artir'p,  to  note  the  inmc  in  the  margin  of  (he  I>ay  Book, 

"V  ^vritiIl^  the  word  F-nlernl,  or  making  two  parallel  stiok'.-s  v.itl.  the  pen  ; 

to  \vhieh  should  be  ad'ii  d  the  figure  denoting  the  page  iu  t'ac  Le£,er,  where 

•nnt  in. 

On  a  blank  p;i£e  at  the  Winning,  or  opd  of  the  Legcr,  an  alphabetic*'. 
>IK!CX  >houliJ  br  \\ritti-n.  ,•  .witaming  the  names  >->f  every  rx-r^i-.i  v  :»h  whona 

'•••i  l-.iv.'  :.,  !-'.;:nt«,  in  ».b«  I.» per.  ivhh  the  number  oi  the  page  v,*:sre  'he 


M  OF  A  LE< . 


Dr. 


| 

C_ 

<>ds, 

ll 

... 

. 

50 

8 

B    i 

1 

Dr. 


Sam;; 


JauY. 


.'>  To  •?  -Vcr  at 

ccuin  a  wi-«?k. 


Dr. 


Anthony  BiiS 


. 

$ 

c. 

Barrel  of  (                    .rrel, 

1 

10 

paid  }  our  onlor  in  Savor  of  G.  Gilbert 

April 

6 



U 



ditto  

-43 

MY  oirlrr  on  Thorn  11  Gro^vcuor, 

54 

«JO 

Dr. 


Thomas  Gro^vciior, 


Jan'y. 

"•ame  of  a  IIOUKP, 

irii'S.       ...... 

1 

FebV 

10 

. 

iVamc  of  a  i»aru,    .... 

April 

Sun;                  ..... 

Dr. 


Edward  Jones, 


24  To  ray  draft  on  Thomas  Gros'venor, 


Dr. 


i- 


185?.    I 

Feb'y.  I  -,'0 1  To  sundries, 


FORM  OF  A  LEGER. 


A  hired  lad,                        Cr. 

188*. 

i 

C. 

Jan'y. 
May 

1 

15 

By  3  months' 
•1*  months' 

wages  due  this  day  at  $G,     . 
wages  at  $7,       .... 

18 
31 

00 

so 

Farmer. 


Cr. 


Merchant, 


Cr. 


1822. 

$ 

C. 

Jan'y. 

Jfl 

By  ray  order  in  favor  of  Joseph  Hastings, 

11 

50 

Sundries,       

SH 

62 

Feb'v. 

•?!! 

ditto  

3 

Sri 

4.pril 

o>0 

ditto.          

c, 

Of» 

Judge  of  County  Court, 


Cr. 


Fy.  V22 

12  By  my  order  iu  favor  of  Joseph  11-as.tiajjs,        | 

24 

My  draft  in  favor  of  Ed^rard  Jontfi, 

00 

March 

4 

Cash  paid  me  this  day,       .       . 

75 

00 

7 

1  empty  cider  barrel,     .... 

58 

April 

I2| 

Amount  of  your  order  on  Theodore  Barrel!, 

68 

00 

May 

25|     My  order  in  favor  of  Anthony  Billings, 

54 

00 

Labourer, 


Cr. 


1822.  |     I 

JanV.  j28jBy  team  hire  at  sumlry  times,       .       .       .        "5J64 

Feb'y.  loj     4  ri>'.'!.1hH'  hire  of  his  son  William  at  $10, 


Far; 


1822. 
March 


1  ;.|By  sundries  in  full. 


Cr. 

TIFc. 

551 


T.r.ER. 


Dr. 


Samuel  Green, 


May    128 


Tocas>'=  count. 


Dr. 


Sept. 


To  cash  paid  for  my  cole  to  D.  Mason, 


Dr. 


Jonathan  Cu 


1822.  I 

Marchpl  To  a  bay  horse, 

I     i 


Dr. 


April   ttSiTo  1  '  .  > 


Dr. 


The od  re  B.irrell, 


May 


3ITol6ri.  .  ibs.  at  5  <  , 

. 
24  Ibs.  h< 


,..OE\  TO  THE  I.EC::K. 

B. 

H, 

FADE 

FA6K. 

Br.rrHl  Theodore, 

IlriMirv 

I 

Billn 

.       I 

! 

1 

Curtis  Jonathan,  . 

Notes  Payable,    . 

.  a 

Daboli  Peter, 

.       1 

R. 

. 

•nor  Thorn 

.       1 

S. 

. 

.   a 

Stacy  Samuel,     . 

.   i 

FORM  v^T  A  J 


New  Londo 


Cr. 


i 

ulries, 


April  [28 


my  note  to  T) ••.:.  a 

eti'i'.  ,      •     .- 

.  -.tl-v 


Dan1 


tic. 

. 
90|00 

".    ^ 

Cr. 


April     18  By  a  coat, 
I     !     A  pr\ir  01 


Sic. 


Cr. 


New  Lor 


18??. 

May 


Cr. 


20  By  caib  in  full, \^.'.    10 


liUESTIO.VS  TO  EXERCISE  THE  STUDENT. 


Jonathan  ' 

Joho  Ko-i- 


>t  f/ic  state  of  the  following 

j  !>ne  Ju-fph  l-1-..stiiv;-!,    .»     .   g 

—  .       ,.    •,        ... 

r   ' » 

• 

• 
1 1  Joho  ;  j.      '  tiO.OO 


Joseph  f  l'i~* 

i     -     'i'V, 

Anthon 
Thoma 

Edn  ? 


A  .FanwrV  BM.  or  Art<>ant. 

All': 

Thomas  Yates,  Esq. 

To  John  Moniington,  Dr. 

last, 

Aprils.     To  :>  barrels  Cider  r.t.  M       .       .       £  10,00 

2()  bushcb  Potatwec,  at     0,J;>  ... 
5.>  Ibs.  Butter,  at  0,17 

JUDJ  G.  itoaofHay, 

Julv  li. 

rda  ol'  AVi.ii.!,  ;.:         4,00  . 


Received  th«  amount. 

JOHN  ArOFxMNtTON. 

IV.  B. — To  prcvont  accidents,  cai-e  shuukl  he  tukcu  not  to  re- 
«eipt  an  accouat  uulil  it  is  pni.l. 

tlal 

' 

Six  months  after  tlato,  I  promi^  to  y.  ' 

order,  (at  my  house,)  One  llumirc'i  Dollars.  -  ,vr«l  iu 

two  yoke  of  "oxen.  J  A  ML  S  IULI 

jj^7-It  is  bt  -t  to  mention  where  : 
wlral  it  is  «;ivun.      U'ithjat.  ihc  l 

Bot  negotiable.  — 

.•?  /  '~JL 

Fifty  tw.,  Dollars  in  full  of  rD  demands.    <Jr.o.  OiKiDUTN. 

.i\r.iiTii  iiv  'I-)1,  in  lull,  wri; 
;. — For  oilier  useful  fonns  s:?c  th.-    \: 


.VY.' 
The  affectionate  Irisfr  j&*)i- 

l»  LTJfh  contribute  to  tlicir  snccc-^  in  \\:<\  \villi  ••'  l^h 
V5  to  tench  them  to  understand  r 

•  •    •  i 

'    i    !«'   !iU'7U 

^/ 

;r).l    b:il:i:,. 

- 

Fir 


^5ft# 


A     000  025  956     4 


V/  . 


